Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
values |
|---|---|---|
Lemma 10.8.1 Let \( X \) be the point graph of a generalized quadrangle of order \( \left( {s, t}\right) \). Then \( X \) is strongly regular with parameters\n\n\[ \left( {\left( {s + 1}\right) \left( {{st} + 1}\right), s\left( {t + 1}\right), s - 1, t + 1}\right) \text{.} \] | Proof. Each point \( P \) of the generalized quadrangle lies on \( t + 1 \) lines of size \( s + 1 \), any two of which have exactly \( P \) in common. Hence \( X \) has valency \( s\left( {t + 1}\right) \). The graph induced by the points collinear with \( P \) consists of \( t + 1 \) vertex-disjoint cliques of size \... | Yes |
Lemma 10.8.2 The eigenvalues of the point graph of a generalized quadrangle of order \( \left( {s, t}\right) \) are \( s\left( {t + 1}\right), s - 1 \), and \( - t - 1 \), with respective multiplicities | Proof. Let \( X \) be the point graph of a generalized quadrangle of order \( \left( {s, t}\right) \) . From Section 10.2, the eigenvalues of \( X \) are its valency \( s\left( {t + 1}\right) \) and the two zeros of the polynomial\n\n\[{x}^{2} - \left( {a - c}\right) x - \left( {k - c}\right) = {x}^{2} - \left( {s - t ... | Yes |
Lemma 10.8.3 If \( \\mathcal{G} \) is a generalized quadrangle of order \( \\left( {s, t}\\right) \) with \( s > 1 \) and \( t > 1 \), then \( s \\leq {t}^{2} \) and \( t \\leq {s}^{2} \) . | Proof. Let \( X \) be the point graph of \( \\mathcal{G} \) . Substituting \( k = s\\left( {t + 1}\\right) ,\\theta = s - 1 \) , and \( \\tau = - t - 1 \) into the second Krein inequality\n\n\\[ \n{\\theta }^{2}\\tau - {2\\theta }{\\tau }^{2} - {\\tau }^{2} - {k\\tau } + k{\\theta }^{2} + {2k\\theta } \\geq 0\n\\]\n\na... | Yes |
Lemma 10.8.4 If a generalized quadrangle of order \( \left( {2, t}\right) \) exists, then \( t \in \) \( \{ 1,2,4\} \) . | Proof. If \( s = 2 \), then \( - t - 1 \) is an eigenvalue of the point graph with multiplicity\n\n\[ \frac{{8t} + 4}{t + 2} = 8 - \frac{12}{t + 2} \]\n\nTherefore, \( t + 2 \) divides 12, which yields that \( t \in \{ 1,2,4,{10}\} \) . The case \( t = {10} \) is excluded by the Krein bound. | Yes |
Lemma 10.9.1 Let \( X \) be a strongly regular graph with parameters\n\n\[ \left( {{6t} + 3,{2t} + 2,1, t + 1}\right) \text{.} \]\n\nThe spectrum of the second subconstituent of \( X \) is\n\n\[ \left\{ {{\left( t + 1\right) }^{\left( 1\right) },{1}^{\left( x\right) },{\left( 1 - t\right) }^{\left( t + 1\right) },{\lef... | Proof. The first subconstituent of \( X \) has valency one, and hence consists of \( t + 1 \) vertex-disjoint edges. Its eigenvalues are 1 and -1, each with multiplicity \( t + 1 \), and so -1 is the unique local eigenvalue of the first subconstituent. Therefore, the nonlocal eigenvalues of the second subconstituent of... | Yes |
Lemma 10.9.2 The graph \( L\left( {K}_{3,3}\right) \) is the unique strongly regular graph with parameters \( \left( {9,4,1,2}\right) \) . | Proof. Let \( X \) be a strongly regular graph with parameters \( \left( {9,4,1,2}\right) \) . Every second subconstituent \( {X}_{2} \) is a connected graph with valency two on four vertices, and so is \( {C}_{4} \) . Every edge of \( {C}_{4} \) lies in a unique one-factor, and so in a unique one-factor with a proper ... | No |
Lemma 10.9.3 The graph \( \overline{L\left( {K}_{6}\right) } \) is the unique strongly regular graph with parameters \( \left( {{15},6,1,3}\right) \) . | Proof. The second subconstituent \( {X}_{2} \) is a connected cubic graph on 8 vertices. By Lemma 10.9.1 we find that its spectrum is symmetric, and therefore \( {X}_{2} \) is bipartite. From this we can see that \( {X}_{2} \) cannot have diameter two, and therefore it has diameter at least three. By considering two ve... | Yes |
Lemma 10.9.4 The complement of the Schläfli graph is the unique strongly regular graph with parameters \( \left( {{27},{10},1,5}\right) \) . | Proof. The second subconstituent \( {X}_{2} \) is a connected graph on 16 vertices with valency 5 . Using Lemma 10.9.1 we find that \( {X}_{2} \) has exactly three eigenvalues, and so is strongly regular with parameters \( \left( {{16},5,0,2}\right) \) . We showed in Section 10.6 that the Clebsch graph is the only stro... | Yes |
Corollary 10.9.5 There is a unique generalized quadrangle of each order \( \left( {2,1}\right) ,\left( {2,2}\right) \), and \( \left( {2,4}\right) \) . | Proof. We have shown that the point graph of a generalized quadrangle of these orders is uniquely determined. Therefore, it will suffice to show that the generalized quadrangle can be recovered from its point graph. If \( X \) is a strongly regular graph with parameters \( \left( {{6t} + 3,{2t} + 2,1, t + 1}\right) \),... | Yes |
Lemma 10.10.1 Let \( \mathcal{D} \) be a quasi-symmetric 2- \( \left( {v, k,\lambda }\right) \) design with intersection numbers \( {\ell }_{1} \) and \( {\ell }_{2} \) . Let \( X \) be the graph with the blocks of \( \mathcal{D} \) as its vertices, and with two blocks adjacent if and only if they have exactly \( {\ell... | Proof. Suppose that \( \mathcal{D} \) has \( b \) blocks and that each point lies in \( r \) blocks. If \( N \) is the \( v \times b \) incidence matrix of \( \mathcal{D} \), then from the results in Section 5.10 we have\n\n\[ N{N}^{T} = \left( {r - \lambda }\right) I + {\lambda J} \]\n\nand\n\n\[ {NJ} = {rJ},\;{N}^{T}... | Yes |
Corollary 10.12.3 Let \( X \) be a graph with binary rank \( {2r} \) . Then \( \chi \left( X\right) \leq \) \( {2}^{r} + 1 \) . | Proof. Duplicating vertices or adding isolated vertices does not alter the chromatic number of a graph. Therefore, we can assume without loss of generality that \( X \) is a reduced graph. Thus it is an induced subgraph of \( \operatorname{Sp}\left( {2r}\right) \) and can be coloured with at most \( {2}^{r} + 1 \) colo... | Yes |
Theorem 11.2.1 (The Absolute Bound) Let \( {X}_{1},\ldots ,{X}_{n} \) be the projections onto a set of \( n \) equiangular lines in \( {\mathbb{R}}^{d} \). Then these matrices form a linearly independent set in the space of symmetric matrices, and consequently \( n \leq \left( \begin{matrix} d + 1 \\ 2 \end{matrix}\rig... | Proof. Let \( \alpha \) be the cosine of the angle between the lines. If \( Y = \mathop{\sum }\limits_{i}{c}_{i}{X}_{i} \), then\n\n\[ \operatorname{tr}\left( {Y}^{2}\right) = \mathop{\sum }\limits_{{i, j}}{c}_{i}{c}_{j}\operatorname{tr}\left( {{X}_{i}{X}_{j}}\right) \]\n\n\[ = \mathop{\sum }\limits_{i}{c}_{i}^{2} + \m... | Yes |
Lemma 11.3.1 Suppose that \( {X}_{1},\ldots ,{X}_{n} \) are the projections onto a set of equiangular lines in \( {\mathbb{R}}^{d} \) and that the cosine of the angle between the lines is \( \alpha \) . If \( I = \mathop{\sum }\limits_{i}{c}_{i}{X}_{i} \), then \( {c}_{i} = d/n \) for all \( i \) and\n\n\[ n = \frac{d ... | Proof. For any \( j \) we have\n\n\[ {X}_{j} = \mathop{\sum }\limits_{i}{c}_{i}{X}_{i}{X}_{j} \]\n\nand so by taking the trace we get\n\n\[ 1 = \operatorname{tr}\left( {X}_{j}\right) = \mathop{\sum }\limits_{i}{c}_{i}\operatorname{tr}\left( {{X}_{i}{X}_{j}}\right) = \left( {1 - {\alpha }^{2}}\right) {c}_{j} + {\alpha }... | Yes |
Lemma 11.4.1 Suppose that there are \( n \) equiangular lines in \( {\mathbb{R}}^{d} \) and that \( \alpha \) is the cosine of the angle between them. If \( {\alpha }^{-2} > d \), then\n\n\[ n \leq \frac{d - d{\alpha }^{2}}{1 - d{\alpha }^{2}} \]\n\nIf \( {X}_{1},\ldots ,{X}_{n} \) are the projections onto these lines,... | Proof. Put\n\n\[ Y \mathrel{\text{:=}} I - \frac{d}{n}\mathop{\sum }\limits_{i}{X}_{i} \]\n\nBecause \( Y \) is symmetric, we have \( \operatorname{tr}\left( {Y}^{2}\right) \geq 0 \), with equality if and only if \( Y = 0 \) . Now,\n\n\[ {Y}^{2} = I - \frac{2d}{n}\mathop{\sum }\limits_{i}{X}_{i} + \frac{{d}^{2}}{{n}^{2... | Yes |
Lemma 11.5.1 If \( X \) is a graph and \( \sigma \) is a subset of \( V\left( X\right) \), then \( S\left( X\right) \) and \( S\left( {X}^{\sigma }\right) \) have the same eigenvalues. | Proof. Let \( D \) be the diagonal matrix with \( {D}_{uu} = - 1 \) if \( u \in \sigma \) and 1 otherwise. Then \( {D}^{2} = I \), so \( D \) is its own inverse. Then\n\n\[ S\left( {X}^{\sigma }\right) = {DS}\left( X\right) D \]\n\nso \( S\left( X\right) \) and \( S\left( {X}^{\sigma }\right) \) are similar and have th... | Yes |
Corollary 11.6.2 A nontrivial regular two-graph has an even number of vertices. | Proof. From the above proof, it follows that \( n = - \left( {{4\theta \tau } + 2\left( {\theta + \tau }\right) + 1}\right) \) . Because both \( {\theta \tau } \) and \( \theta + \tau \) are integers, this shows that \( n \) is odd; hence \( n + 1 \) is even. | Yes |
Theorem 11.7.1 Let \( X \) be a \( k \) -regular graph on \( n \) vertices not switching equivalent to the complete or empty graph. Then \( S\left( X\right) \) has two eigenvalues if and only if \( X \) is strongly regular and \( k - n/2 \) is an eigenvalue of \( A\left( X\right) \) . | Proof. Any eigenvector of \( A\left( X\right) \) orthogonal to 1 with eigenvalue \( \theta \) is an eigenvector of \( S\left( X\right) \) with eigenvalue \( - {2\theta } - 1 \), while 1 itself is an eigenvector of \( S\left( X\right) \) with eigenvalue \( n - 1 - {2k} \) . Therefore, if \( X \) is strongly regular with... | Yes |
Theorem 12.2.1 A maximal set of lines at \( {60}^{ \circ } \) and \( {90}^{ \circ } \) in \( {\mathbb{R}}^{n} \) is star-closed. | Proof. Let \( \mathcal{L} \) be a set of lines at \( {60}^{ \circ } \) and \( {90}^{ \circ } \), and suppose that \( \langle a\rangle \) , \( \langle b\rangle \in \mathcal{L} \) are two lines at \( {60}^{ \circ } \) . We can assume that \( a \) and \( b \) have length \( \sqrt{2} \) and choose \( b \) such that \( \lan... | Yes |
Lemma 12.3.1 Let \( \mathcal{L} \) be a set of lines at \( {60}^{ \circ } \) and \( {90}^{ \circ } \) in \( {\mathbb{R}}^{n} \). Then \( \mathcal{L} \) is star-closed if and only if for every vector \( h \) that spans a line in \( \mathcal{L} \), the reflection \( {\rho }_{h} \) fixes \( \mathcal{L} \). | Proof. Let \( h \) be a vector of length \( \sqrt{2} \) spanning a line in \( \mathcal{L} \). From our comments above, \( {\rho }_{h} \) fixes \( \langle h\rangle \) and all the lines orthogonal to \( \langle h\rangle \). So suppose that \( \langle a\rangle \) is a line of \( \mathcal{L} \) at \( {60}^{ \circ } \) to \... | Yes |
Lemma 12.4.1 For \( n \geq 2 \), the set of lines \( \mathcal{L} \) spanned by the vectors in \( {D}_{n} \) is indecomposable. | Proof. The lines \( \left\langle {{e}_{1} + {e}_{i}}\right\rangle \) for \( i \geq 2 \) have pairwise inner products equal to 1, and hence must be in the same part of any decomposition of \( \mathcal{L} \) . It is clear, however, that any other vector in \( {D}_{n} \) has nonzero inner product with at least one of thes... | Yes |
Theorem 12.4.2 Let \( \mathcal{L} \) be a star-closed indecomposable set of lines at \( {60}^{ \circ } \) and \( {90}^{ \circ } \). Then the reflection group of \( \mathcal{L} \) acts transitively on ordered pairs of nonorthogonal lines. | Proof. First we observe that the reflection group acts transitively on the lines of \( \mathcal{L} \). Suppose that \( \langle a\rangle \) and \( \langle b\rangle \) are two lines that are not orthogonal, and that \( \langle a, b\rangle = - 1 \). Then \( c = - a - b \) spans the third line in the star with \( \langle a... | Yes |
Lemma 12.4.3 If \( X \) is a graph with minimum eigenvalue at least -2, then the star-closed set of lines \( \mathcal{L}\left( X\right) \) is indecomposable if and only if \( X \) is connected. | Proof. First suppose that \( X \) is connected. Let \( {\mathcal{L}}^{\prime } \) be the lines spanned by the vectors whose Gram matrix is \( A\left( X\right) + {2I} \) . Lines corresponding to adjacent vertices of \( X \) are not orthogonal, and hence must be in the same part of any decomposition of \( \mathcal{L}\lef... | Yes |
Lemma 12.5.1 Let \( \mathcal{L} \) be an indecomposable star-closed set of lines at \( {60}^{ \circ } \) and \( {90}^{ \circ } \), and let \( \langle a\rangle ,\langle b\rangle \), and \( \langle c\rangle \) form a star in \( \mathcal{L} \). Every other line of \( \mathcal{L} \) is orthogonal to either one or three lin... | Proof. Without loss of generality we may assume that \( a, b \), and \( c \) all have length \( \sqrt{2} \) and that\n\n\[ \langle a, b\rangle = \langle b, c\rangle = \langle c, a\rangle = - 1.\]\n\nIt follows then that \( c = - a - b \), and so for any other line \( \langle x\rangle \) of \( \mathcal{L} \) we have\n\n... | Yes |
Lemma 12.5.2 The set \( \mathcal{L} \) is the star-closure of \( \langle a\rangle ,\langle b\rangle \), and \( C \) . | Proof. Let \( \mathcal{M} \) denote the set of lines \( \{ \langle a\rangle ,\langle b\rangle \} \cup C \) . Clearly, \( \langle c\rangle \) lies in the star-closure of \( \mathcal{M} \), and so it suffices to show that every line in \( A, B \), and \( D \) lies in a star with two lines chosen from \( \mathcal{M} \) . ... | Yes |
Lemma 12.6.1 If \( x \) and \( y \) are orthogonal vectors in \( {C}^{ * } \), then there is a unique vector in \( {C}^{ * } \) orthogonal to both of them. | Proof. Suppose that vectors \( x, y \in {C}^{ * } \) are orthogonal. Then by our comments above, we see that \( x + b \in {A}^{ * } \) and that \( y - a \in {B}^{ * } \) and that \( \langle x + b, y - a\rangle = - 1 \) . Therefore, \( \langle a - b - x - y\rangle \in \mathcal{L} \), and calculation shows that \( a - b ... | Yes |
Theorem 12.6.2 Let \( \mathcal{Q} \) be the incidence structure whose points are the vectors of \( {C}^{ * } \), and whose lines are triples of mutually orthogonal vectors. Then either \( \mathcal{Q} \) has no lines, or \( \mathcal{Q} \) is a generalized quadrangle, possibly degenerate, with lines of size three. | Proof. A generalized quadrangle has the property that given any line \( \ell \) and a point \( P \) off that line, there is a unique point on \( \ell \) collinear with \( P \) . We show that \( \mathcal{Q} \) satisfies this axiom.\n\nSuppose that \( x, y \), and \( a - b - x - y \) are the three points of a line of \( ... | Yes |
Theorem 12.7.2 The root system \( {E}_{8} \) contains exactly 240 vectors. The lines spanned by these vectors form an indecomposable star-closed set of lines at \( {60}^{ \circ } \) and \( {90}^{ \circ } \) in \( {\mathbb{R}}^{8} \) . The generalized quadrangle \( \mathcal{Q} \) associated with this set of lines is the... | Proof. This is immediate, since \( {D}_{8} \) contains 112 vectors, and there are 128 further vectors. | No |
Theorem 12.7.4 An indecomposable star-closed set of lines at \( {60}^{ \circ } \) and \( {90}^{ \circ } \) is the set of lines spanned by the vectors in one of the root systems \( {E}_{6} \) , \( {E}_{7},{E}_{8},{A}_{n} \), or \( {D}_{n} \) (for some \( n \) ). | Proof. The Gram matrix of the vectors in \( {C}^{ * } \) determines the Gram matrix of the entire collection of lines in \( \mathcal{L} \), which in turn determines \( \mathcal{L} \) up to an orthogonal transformation. Since these five root systems give the only five possible Gram matrices for the vectors in \( {C}^{ *... | Yes |
Theorem 12.7.4 An indecomposable star-closed set of lines at \( {60}^{ \circ } \) and \( {90}^{ \circ } \) is the set of lines spanned by the vectors in one of the root systems \( {E}_{6} \) , \( {E}_{7},{E}_{8},{A}_{n} \), or \( {D}_{n} \) (for some \( n \) ). | Proof. The Gram matrix of the vectors in \( {C}^{ * } \) determines the Gram matrix of the entire collection of lines in \( \mathcal{L} \), which in turn determines \( \mathcal{L} \) up to an orthogonal transformation. Since these five root systems give the only five possible Gram matrices for the vectors in \( {C}^{ *... | Yes |
Corollary 12.8.1 Let \( X \) be a connected graph with smallest eigenvalue at least -2, and let \( A \) be its adjacency matrix. Then either \( X \) is a generalized line graph, or \( A + {2I} \) is the Gram matrix of a set of vectors in \( {E}_{8} \) . | Proof. Let \( S \) be a set of vectors with Gram matrix \( {2I} + A \) . Then the star-closure of \( S \) is contained in the set of lines spanned by the vectors in \( {E}_{8} \) or \( {D}_{n} \) . | No |
Theorem 12.8.2 Let \( X \) be a graph with least eigenvalue at least -2 . If \( X \) has more than 36 vertices or maximum valency greater than 28, it is a generalized line graph. | Proof. If \( X \) is not a generalized line graph, then \( A\left( X\right) + {2I} \) is the Gram matrix of a set of vectors in \( {E}_{8} \) . So let \( S \) be a set of vectors from \( {E}_{8} \) with nonnegative pairwise inner products. First we will show that \( \left| S\right| \leq {36} \) . For any vector \( x \i... | Yes |
Lemma 13.1.2 Let \( X \) be a regular graph with valency \( k \) . If the adjacency matrix \( A \) has eigenvalues \( {\theta }_{1},\ldots ,{\theta }_{n} \), then the Laplacian \( Q \) has eigenvalues \( k - {\theta }_{1},\ldots, k - {\theta }_{n} \) . | Proof. If \( X \) is \( k \) -regular, then \( Q = \Delta \left( X\right) - A = {kI} - A \) . Thus every eigenvector of \( A \) with eigenvalue \( \theta \) is an eigenvector of \( Q \) with eigenvalue \( k - \theta \) . | Yes |
Lemma 13.1.3 If \( X \) is a graph on \( n \) vertices and \( 2 \leq i \leq n \), then \( {\lambda }_{i}\left( \bar{X}\right) = \) \( n - {\lambda }_{n - i + 2}\left( X\right) \) . | Proof. We start by observing that\n\n\[ Q\left( X\right) + Q\left( \bar{X}\right) = {nI} - J. \]\n\n(13.1)\n\nThe vector \( \mathbf{1} \) is an eigenvector of \( Q\left( X\right) \) and \( Q\left( \bar{X}\right) \) with eigenvalue 0 . Let \( x \) be another eigenvector of \( Q\left( X\right) \) with eigenvalue \( \lamb... | Yes |
Lemma 13.1.5 Let \( X \) be a graph on \( n \) vertices with Laplacian \( Q \) . Then for any vector \( x \) ,\n\n\[ \n{x}^{T}{Qx} = \mathop{\sum }\limits_{{{uv} \in E\left( X\right) }}{\left( {x}_{u} - {x}_{v}\right) }^{2}.\n\] | Proof. This follows from the observations that\n\n\[ \n{x}^{T}{Qx} = {x}^{T}D{D}^{T}x = {\left( {D}^{T}x\right) }^{T}\left( {{D}^{T}x}\right)\n\]\n\nand that if \( {uv} \in E\left( X\right) \), then the entry of \( {D}^{T}x \) corresponding to \( {uv} \) is \( \pm \left( {{x}_{u} - {x}_{v}}\right) \) . | No |
Theorem 13.2.1 Let \( X \) be a graph with Laplacian matrix \( Q \) . If \( u \) is an arbitrary vertex of \( X \), then \( \det Q\left\lbrack u\right\rbrack \) is equal to the number of spanning trees of \( X \) . | Proof. We prove the theorem by induction on the number of edges of \( X \) . Let \( \tau \left( X\right) \) denote the number of spanning trees of \( X \) . If \( e \) is an edge of \( X \), then every spanning tree either contains \( e \) or does not contain \( e \) , so we can count them according to this distinction... | Yes |
Corollary 13.2.2 The number of spanning trees of \( {K}_{n} \) is \( {n}^{n - 2} \) . | Proof. This follows directly from the fact that \( Q\left\lbrack u\right\rbrack = n{I}_{n - 1} - J \) for any vertex \( u \) . | No |
Lemma 13.2.3 Let \( \tau \left( X\right) \) denote the number of spanning trees in the graph \( X \) and let \( Q \) be its Laplacian. Then \( \operatorname{adj}\left( Q\right) = \tau \left( X\right) J \) . | Proof. Suppose that \( X \) has \( n \) vertices. Assume first that \( X \) is not connected, so that \( \tau \left( X\right) = 0 \) . Then \( Q \) has rank at most \( n - 2 \), so any submatrix of \( Q \) of order \( \left( {n - 1}\right) \times \left( {n - 1}\right) \) is singular and \( \operatorname{adj}\left( Q\ri... | Yes |
Lemma 13.2.4 Let \( X \) be a graph on \( n \) vertices, and let \( {\lambda }_{1},\ldots ,{\lambda }_{n} \) be the eigenvalues of the Laplacian of \( X \) . Then the number of spanning trees in \( X \) is \( \frac{1}{n}\mathop{\prod }\limits_{{i = 2}}^{n}{\lambda }_{i} \) . | Proof. The result clearly holds if \( X \) is not connected, so we may assume without loss that \( X \) is connected. Let \( \phi \left( t\right) \) denote the characteristic polynomial \( \det \left( {{tI} - Q}\right) \) of the Laplacian \( Q \) of \( X \) . The zeros of \( \phi \left( t\right) \) are the eigenvalues ... | Yes |
Lemma 13.3.1 Let \( \\rho \) be a representation of the edge-weighted graph \( X \) , given by the \( \\left| {V\\left( X\\right) }\\right| \\times m \) matrix \( R \) . If \( D \) is an oriented incidence matrix for \( X \), then\n\n\[\\mathcal{E}\\left( \\rho \\right) = \\operatorname{tr}{R}^{T}{DW}{D}^{T}R\] | Proof. The rows of \( {D}^{T}R \) are indexed by the edges of \( X \), and if \( {uv} \\in E\\left( X\\right) \) , then the \( {uv} \)-row of \( {D}^{T}R \) is \( \\pm \\left( {\\rho \\left( u\\right) - \\rho \\left( v\\right) }\\right) \) . Consequently, the diagonal entries of \( {D}^{T}R{R}^{T}D \) have the form \( ... | Yes |
Theorem 13.4.1 Let \( X \) be a graph on \( n \) vertices with weighted Laplacian Q. Assume that the eigenvalues of \( Q \) are \( {\lambda }_{1} \leq \cdots \leq {\lambda }_{n} \) and that \( {\lambda }_{2} > 0 \) . The minimum energy of a balanced orthogonal representation of \( X \) in \( {\mathbb{R}}^{m} \) equals ... | Proof. By Lemma 13.3.1 the energy of a representation is \( \operatorname{tr}{R}^{T}{QR} \) . From Corollary 9.5.2, the energy of an orthogonal representation in \( {\mathbb{R}}^{\ell } \) is bounded below by the sum of the \( \ell \) smallest eigenvalues of \( Q \) . We can realize this lower bound by taking the colum... | Yes |
Theorem 13.5.1 Suppose that \( S \) is a subset of the vertices of the graph \( X \) . Then \( {\lambda }_{2}\left( X\right) \leq {\lambda }_{2}\left( {X \smallsetminus S}\right) + \left| S\right| \) . | Proof. Let \( z \) be a unit vector of length \( n \) such that (when viewed as a function on \( V\left( X\right) \) ) its restriction to \( S \) is zero, and its restriction to \( V\left( X\right) \smallsetminus S \) is an eigenvector of \( Q\left( {X \smallsetminus S}\right) \) orthogonal to 1 and with eigenvalue \( ... | Yes |
Corollary 13.5.2 For any graph \( X \) we have \( {\lambda }_{2}\left( X\right) \leq {\kappa }_{0}\left( X\right) \) . | It follows from our observation in Section 13.1 or from Exercise 4 that the characteristic polynomial of \( Q\left( {K}_{1, n}\right) \) is \( t{\left( t - 1\right) }^{n - 1}\left( {t - n - 1}\right) \) . This provides one family of examples where \( {\lambda }_{2} \) equals the vertex connectivity.\n\nProvided that \(... | Yes |
Lemma 13.6.1 Let \( X \) be a graph and let \( Y \) be obtained from \( X \) by adding an edge joining two distinct vertices of \( X \) . Then\n\n\[ \n{\lambda }_{2}\left( X\right) \leq {\lambda }_{2}\left( Y\right) \leq {\lambda }_{2}\left( X\right) + 2 \n\] | Proof. Suppose we get \( Y \) by joining vertices \( r \) and \( s \) of \( X \) . For any vector \( z \) we have\n\n\[ \n{z}^{T}Q\left( Y\right) z = \mathop{\sum }\limits_{{{uv} \in E\left( Y\right) }}{\left( {z}_{u} - {z}_{v}\right) }^{2} = {\left( {z}_{r} - {z}_{s}\right) }^{2} + \mathop{\sum }\limits_{{{uv} \in E\l... | Yes |
Theorem 13.6.2 Let \( X \) be a graph with \( n \) vertices and let \( Y \) be obtained from \( X \) by adding an edge joining two distinct vertices of \( X \) . Then \( {\lambda }_{i}\left( X\right) \leq \) \( {\lambda }_{i}\left( Y\right) \), for all \( i \), and \( {\lambda }_{i}\left( Y\right) \leq {\lambda }_{i + ... | Proof. Suppose we add the edge \( {uv} \) to \( X \) to get \( Y \) . Let \( z \) be the vector of length \( n \) with \( u \) -entry and \( v \) -entry 1 and -1, respectively, and all other entries equal to 0 . Then \( Q\left( Y\right) = Q\left( X\right) + z{z}^{T} \), and if we use \( Q \) to denote \( Q\left( X\righ... | Yes |
Lemma 13.7.1 Let \( X \) be a graph on \( n \) vertices and let \( S \) be a subset of \( V\left( X\right) \) . Then \[ {\lambda }_{2}\left( X\right) \leq \frac{n\left| {\partial S}\right| }{\left| S\right| \left( {n - \left| S\right| }\right) } \] | Proof. Suppose \( \left| S\right| = a \) . Let \( z \) be the vector (viewed as a function on \( V\left( X\right) ) \) whose value is \( n - a \) on the vertices in \( S \) and \( - a \) on the vertices not in \( S \) . Then \( z \) is orthogonal to 1, so by Corollary 13.4.2 \[ {\lambda }_{2}\left( X\right) \leq \frac{... | Yes |
Corollary 13.7.3 The bisection width of a graph \( X \) on \( {2m} \) vertices is at least \( m{\lambda }_{2}\left( X\right) /2 \) . | We apply this to the \( k \) -cube \( {Q}_{k} \) . In Exercise 13 it is established that \( {\lambda }_{2}\left( {Q}_{k}\right) = 2 \), from which it follows that the bisection width of the \( k \) -cube is at least \( {2}^{k - 1} \) . Since this value is easily realized, we have thus found the exact value. | No |
Lemma 13.7.4 If \( X \) is a graph with \( n \) vertices, then \( \operatorname{bip}\left( X\right) \leq n{\lambda }_{\infty }\left( X\right) /4 \) . | Proof. By applying Lemma 13.7.1 to the complement of \( X \) we get\n\n\[ \left| {\partial S}\right| \leq \left| S\right| \left( {n - \left| S\right| }\right) {\lambda }_{\infty }\left( X\right) /n \leq n{\lambda }_{\infty }\left( X\right) /4 \]\n\nwhich is the desired inequality. | Yes |
Lemma 13.8.1 Let \( S \) be a set of points in \( {\mathbb{R}}^{m} \). Then the vector \( x \) in \( {\mathbb{R}}^{m} \) minimizes \( \mathop{\sum }\limits_{{y \in S}}\parallel x - y{\parallel }^{2} \) if and only if | Proof. Let \( \widehat{y} \) be the centroid of the set \( S \), i.e., \[ \widehat{y} = \frac{1}{\left| S\right| }\mathop{\sum }\limits_{{y \in S}}y \] Then \[ \mathop{\sum }\limits_{{y \in S}}\parallel x - y{\parallel }^{2} = \mathop{\sum }\limits_{{y \in S}}\parallel \left( {x - \widehat{y}}\right) + \left( {\widehat... | Yes |
Lemma 13.8.2 Let \( F \) be a subset of the vertices of \( X \), let \( \rho \) be a representation of \( X \), and let \( R \) be the matrix whose rows are the images of the vertices of \( X \). Let \( Q \) be the Laplacian of \( X \). Then \( \rho \) is barycentric relative to \( F \) if and only if the rows of \( {Q... | Proof. The vector \( x \) is the centroid of the vectors in \( S \) if and only if\n\n\[ \mathop{\sum }\limits_{{y \in S}}\left( {x - y}\right) = 0 \]\n\nIf \( u \) has valency \( d \), the \( u \) -row of \( {QR} \) is equal to\n\n\[ {d\rho }\left( u\right) - \mathop{\sum }\limits_{{v \sim u}}\rho \left( v\right) = \m... | Yes |
Lemma 13.8.3 Let \( X \) be a connected graph, let \( F \) be a subset of the vertices of \( X \), and let \( \sigma \) be a map from \( F \) into \( {\mathbb{R}}^{m} \). If \( X \smallsetminus F \) is connected, there is a unique \( m \) -dimensional representation \( \rho \) of \( X \) that extends \( \sigma \) and i... | Proof. Let \( Q \) be the Laplacian of \( X \). Assume that we have\n\n\[ Q = \left( \begin{matrix} {Q}_{1} & {B}^{T} \\ B & {Q}_{2} \end{matrix}\right) \]\n\nwhere the rows and columns of \( {Q}_{1} \) are indexed by the vertices of \( F \). Let \( R \) be the matrix describing the representation \( \rho \). We may as... | Yes |
Lemma 13.9.1 Let \( X \) be a graph with a generalized Laplacian \( Q \) . If \( X \) is connected, then \( {\lambda }_{1}\left( Q\right) \) is simple and the corresponding eigenvector can be taken to have all its entries positive. | Proof. Choose a constant \( c \) such that all diagonal entries of \( Q - {cI} \) are nonpositive. By the Perron-Frobenius theorem (Theorem 8.8.1), the largest eigenvalue of \( - Q + {cI} \) is simple and the associated eigenvector may be taken to have only positive entries. | Yes |
Lemma 13.9.2 Let \( x \) be an eigenvector of \( Q \) with eigenvalue \( \lambda \) and let \( Y \) be a positive nodal domain of \( x \) . Then \( \left( {Q - {\lambda I}}\right) {x}_{Y} \leq 0 \) . | Proof. Let \( y \) denote the restriction of \( x \) to \( V\left( Y\right) \) and let \( z \) be the restriction of \( x \) to \( V\left( X\right) \smallsetminus {\operatorname{supp}}_{ + }\left( x\right) \) . Let \( {Q}_{Y} \) be the submatrix of \( Q \) with rows and columns indexed by \( V\left( Y\right) \), and le... | Yes |
Corollary 13.9.3 Let \( x \) be an eigenvector of \( Q \) with eigenvalue \( \lambda \), and let \( U \) be the subspace spanned by the vectors \( {x}_{Y} \), where \( Y \) ranges over the positive nodal domains of \( x \) . If \( u \in U \), then \( {u}^{T}\left( {Q - {\lambda I}}\right) u \leq 0 \) . | Proof. If \( u = \mathop{\sum }\limits_{Y}{a}_{Y}{x}_{Y} \), then using (13.6), we find that\n\n\[ \n{u}^{T}\left( {Q - {\lambda I}}\right) u = \mathop{\sum }\limits_{Y}{a}_{Y}^{2}{x}_{Y}^{T}\left( {Q - {\lambda I}}\right) {x}_{Y} \n\]\n\nand so the claim follows from the previous lemma. | Yes |
Theorem 13.9.4 Let \( X \) be a connected graph, let \( Q \) be a generalized Laplacian of \( X \), and let \( x \) be an eigenvector for \( Q \) with eigenvalue \( {\lambda }_{2}\left( Q\right) \) . If \( x \) has minimal support, then \( {\operatorname{supp}}_{ + }\left( x\right) \) and \( {\operatorname{supp}}_{ - }... | Proof. Suppose that \( v \) is a \( {\lambda }_{2} \) -eigenvector with distinct positive nodal domains \( Y \) and \( Z \) . Because \( X \) is connected, \( {\lambda }_{1} \) is simple and the span of \( {v}_{Y} \) and \( {v}_{Z} \) contains a vector, \( u \) say, orthogonal to the \( {\lambda }_{1} \) -eigenspace.\n... | Yes |
Lemma 13.9.5 Let \( Q \) be a generalized Laplacian of a graph \( X \) and let \( x \) be an eigenvector of \( Q \) . Then any vertex not in \( \operatorname{supp}\left( x\right) \) either has no neighbours in \( \operatorname{supp}\left( x\right) \), or has neighbours in both \( {\operatorname{supp}}_{ + }\left( x\rig... | Proof. Suppose that \( u \notin \operatorname{supp}\left( x\right) \), so \( {x}_{u} = 0 \) . Then\n\n\[ 0 = {\left( Qx\right) }_{u} = {Q}_{uu}{x}_{u} + \mathop{\sum }\limits_{{v \sim u}}{Q}_{uv}{x}_{v} = \mathop{\sum }\limits_{{v \sim u}}{Q}_{uv}{x}_{v}. \]\n\nSince \( {Q}_{uv} < 0 \) when \( v \) is adjacent to \( u ... | Yes |
Lemma 13.10.1 Let \( Q \) be a generalized Laplacian for the graph \( X \) . If \( X \) is 3-connected and planar, then no eigenvector of \( Q \) with eigenvalue \( {\lambda }_{2}\left( Q\right) \) vanishes on three vertices in the same face of any embedding of \( X \) . | Proof. Let \( x \) be an eigenvector of \( Q \) with eigenvalue \( {\lambda }_{2} \), and suppose that \( u, v \), and \( w \) are three vertices not in \( \operatorname{supp}\left( x\right) \) lying in the same face. We may assume that \( x \) has minimal support, and hence \( {\operatorname{supp}}_{ + }\left( x\right... | Yes |
Corollary 13.10.2 Let \( Q \) be a generalized Laplacian for the graph \( X \) . If \( X \) is 3-connected and planar, then \( {\lambda }_{2}\left( Q\right) \) has multiplicity at most three. | Proof. If \( {\lambda }_{2} \) has multiplicity at least four, then there is an eigenvector in the associated eigenspace whose support is disjoint from any three given vertices. Thus we conclude that \( {\lambda }_{2} \) has multiplicity at most three. | No |
Lemma 13.10.3 Let \( X \) be a 2-connected plane graph with a generalized Laplacian \( Q \), and let \( x \) be an eigenvector of \( Q \) with eigenvalue \( {\lambda }_{2}\left( Q\right) \) and with minimal support. If \( u \) and \( v \) are adjacent vertices of a face \( F \) such that \( {x}_{u} = {x}_{v} = 0 \), th... | Proof. Since \( X \) is 2-connected, the face \( F \) is a cycle. Suppose that \( F \) contains vertices \( p \) and \( q \) such that \( {x}_{p} > 0 \) and \( {x}_{q} < 0 \) . Without loss of generality we can assume that they occur in the order \( u, v, q \), and \( p \) clockwise around the face \( F \), and that th... | Yes |
Corollary 13.10.4 Let \( X \) be a graph on \( n \) vertices with a generalized Laplacian \( Q \) . If \( X \) is 2-connected and outerplanar, then \( {\lambda }_{2}\left( Q\right) \) has multiplicity at most two. | Proof. If \( {\lambda }_{2} \) had multiplicity greater than two, then we could find an eigenvector \( x \) with eigenvalue \( {\lambda }_{2} \) such that \( x \) vanished on two adjacent vertices in the sole face of \( X \) . However, since \( x \) must be orthogonal to the eigenvector with eigenvalue \( {\lambda }_{1... | Yes |
Lemma 13.11.1 Let \( X \) be a 3-connected planar graph with a generalized Laplacian \( Q \) such that \( {\lambda }_{2}\left( Q\right) \) has multiplicity three. Let \( \rho \) be a representation given by a matrix \( U \) whose columns form a basis for the \( {\lambda }_{2} \) -eigenspace of \( Q \) . If \( F \) is a... | Proof. Assume by way of contradiction that \( u \) and \( v \) are two vertices in a face of \( X \) such that \( \rho \left( u\right) = {\alpha \rho }\left( v\right) \) for some real number \( \alpha \), and let \( w \) be a third vertex in the same face. Then we can find a linear combination of the columns of \( U \)... | Yes |
Lemma 13.11.2 Let \( X \) be a 2-connected planar graph. Suppose it has a planar embedding where the neighbours of the vertex \( u \) are, in cyclic order, \( {v}_{1},\ldots ,{v}_{k} \) . Let \( Q \) be a generalized Laplacian for \( X \) such that \( {\lambda }_{2}\left( Q\right) \) has multiplicity three. Then the pl... | Proof. Let \( x \) be an eigenvector with eigenvalue \( {\lambda }_{2} \) with minimal support such that \( x\left( u\right) = x\left( {v}_{1}\right) = 0 \) . (Here we are viewing \( x \) as a function on \( V\left( X\right) \) .) By Lemma 13.10.1, we see that neither \( x\left( {v}_{2}\right) \) nor \( x\left( {v}_{k}... | Yes |
Lemma 14.1.2 If \( X \) is a graph, then the signed characteristic vector of each cut lies in the cut space of \( X \) . The nonzero elements of the cut space with minimal support are scalar multiples of the signed characteristic vectors of the bonds of \( X \) . | Proof. First let \( C \) be a cut in \( X \) and suppose that \( V\left( +\right) \) and \( V\left( -\right) \) are its shores. Let \( y \) be the characteristic vector in \( {\mathbb{R}}^{V} \) of \( V\left( +\right) \) and consider the vector \( {D}^{T}y \) . It takes the value 0 on any edge with both ends in the sam... | Yes |
Lemma 14.1.3 Let \( X \) be a connected graph and let \( T \) be a spanning tree of \( X \) . Then the signed characteristic vectors of the \( n - 1 \) cuts \( C\left( {T, e}\right) \), for \( e \in E\left( T\right) \), form a basis for the cut space of \( X \) . | Proof. An edge \( e \in E\left( T\right) \) is in the cut \( C\left( {T, e}\right) \) but not in any of the cuts \( C\left( {T, f}\right) \) for \( f \neq e \) . Therefore, the signed characteristic vectors of the cuts are linearly independent, and since there are \( n - 1 \) vectors, they form a basis. | Yes |
Theorem 14.2.2 If \( X \) is a graph, then the signed characteristic vector of each cycle lies in the flow space of \( X \) . The nonzero elements of the flow space with minimal support are scalar multiples of the signed characteristic vectors of the cycles of \( X \) . | Proof. If \( C \) is a cycle with signed characteristic vector \( z \), then it is a straightforward exercise to verify that \( {Dz} = 0 \) .\n\nSuppose, then, that \( y \) lies in the flow space of \( X \) and that its support is minimal. Let \( Y \) denote the subgraph of \( X \) formed by the edges in \( \operatorna... | No |
Corollary 14.2.3 The flow space of \( X \) is spanned by the signed characteristic vectors of its cycles. | There are also a number of natural bases for the flow space of a graph. If \( F \) is a maximal spanning forest of \( X \), then any edge not in \( F \) is called a chord of \( F \) . If \( e \) is a chord of \( F \), then \( e \) together with the path in \( F \) from the head of \( e \) to the tail of \( e \) is a cy... | Yes |
Theorem 14.2.4 Let \( X \) be a graph with \( n \) vertices, \( m \) edges, and \( c \) connected components. Suppose that the rows of the \( \left( {n - c}\right) \times e \) matrix\n\n\[ M = \left( \begin{array}{ll} I & R \end{array}\right) \]\n\nform a basis for the cut space of \( X \) . Then the rows of the \( \le... | Proof. It is obvious that \( M{N}^{T} = 0 \) . Therefore, the rows of \( N \) are in the flow space of \( X \), and since they are linearly independent, they form a basis for the flow space of \( X \) . | Yes |
Theorem 14.3.1 If \( X \) is a plane graph, then a set of edges is a cycle in \( X \) if and only if it is a bond in the dual graph \( {X}^{ * } \) . | Proof. We shall show that a set of edges \( D \subseteq E\left( X\right) \) contains a cycle of \( X \) if and only if it contains a bond of \( {X}^{ * } \) (here we are identifying the edges of \( X \) and \( {X}^{ * } \) ). If \( D \) contains a cycle \( C \), then this forms a closed curve in the plane, and every fa... | Yes |
Lemma 14.3.3 If \( X \) is a connected plane graph and \( T \) a spanning tree of \( X \), then \( E\left( X\right) \smallsetminus E\left( T\right) \) is a spanning tree of \( {X}^{ * } \) . | Proof. The tree \( T \) contains no cycle of \( X \), and therefore \( T \) contains no bond of \( {X}^{ * } \) . Therefore, the graph with vertex set \( V\left( {X}^{ * }\right) \) and edge set \( E\left( X\right) \smallsetminus E\left( T\right) \) is connected. Euler’s formula shows that \( \left| {E\left( X\right) \... | Yes |
Lemma 14.3.3 If \( X \) is a connected plane graph and \( T \) a spanning tree of \( X \), then \( E\left( X\right) \smallsetminus E\left( T\right) \) is a spanning tree of \( {X}^{ * } \) . | Proof. The tree \( T \) contains no cycle of \( X \), and therefore \( T \) contains no bond of \( {X}^{ * } \) . Therefore, the graph with vertex set \( V\left( {X}^{ * }\right) \) and edge set \( E\left( X\right) \smallsetminus E\left( T\right) \) is connected. Euler’s formula shows that \( \left| {E\left( X\right) \... | Yes |
Lemma 14.4.1 Let \( Y \) be a graph with no cut-edges and let \( X \) be a graph obtained by adding an ear to \( Y \) . Then \( X \) has no cut-edges. | Proof. Assume that \( X \) is obtained from \( Y \) by adding a path \( P \) and suppose \( e \in E\left( X\right) \) . If \( e \in E\left( P\right) \), then \( X \smallsetminus e \) is connected. If \( e \in Y \), then \( Y \smallsetminus e \) is connected (because \( Y \) has no cut-edges), and hence \( X \smallsetmi... | Yes |
Theorem 14.4.2 A connected graph \( X \) has an ear decomposition if and only if it has no cut-edges. | Proof. It remains only to prove that \( X \) has an ear decomposition if it has no cut-edges. In fact, we will prove something slightly stronger, which is that \( X \) has an ear decomposition starting with any cycle. Let \( {Y}_{0} \) be any cycle of \( X \) and form a sequence of graphs \( {Y}_{0},{Y}_{1},\ldots \) a... | Yes |
Lemma 14.5.1 The set of all integer linear combinations of a set of linearly independent vectors in \( V \) is a lattice. | Proof. Suppose that \( M \) is a matrix with linearly independent columns. It will be enough to show that there is a positive constant \( \epsilon \) such that if \( y \) is a nonzero integer vector, then \( {y}^{T}{M}^{T}{My} \geq \epsilon \) . But if \( y \) is an integer vector, then \( {y}^{T}y \geq 1 \) . Since \(... | Yes |
Lemma 14.6.1 If the columns of the matrix \( M \) form an integral basis for the lattice \( \mathcal{L} \), then the columns of \( M{\left( {M}^{T}M\right) }^{-1} \) are an integral basis for its dual, \( {\mathcal{L}}^{ * } \) . | Proof. Let \( {a}_{1},\ldots ,{a}_{r} \) denote the columns of \( M \) and \( {b}_{1},\ldots ,{b}_{r} \) the columns of \( M{\left( {M}^{T}M\right) }^{-1} \) . Clearly, the vectors \( {b}_{1},\ldots ,{b}_{r} \) lie in the column space of \( M \), and because \( {M}^{T}M{\left( {M}^{T}M\right) }^{-1} = I \) we have\n\n\... | Yes |
Theorem 14.6.2 If \( M \) is a matrix with linearly independent columns, then projection onto the column space of \( M \) is given by the matrix\n\n\[ P = M{\left( {M}^{T}M\right) }^{-1}{M}^{T} \] | This matrix has the properties that \( P = {P}^{T} \) and \( {P}^{2} = P \) . | No |
Theorem 14.6.3 Suppose that \( M \) is an \( n \times r \) matrix whose columns form an integral basis for the lattice \( \mathcal{L} \) . Let \( P \) be the matrix representing orthogonal projection from \( {\mathbb{R}}^{n} \) onto the column space of \( M \) . If the greatest common divisor of the \( r \times r \) mi... | Proof. From Theorem 14.6.2 we have that \( P = M{\left( {M}^{T}M\right) }^{-1}{M}^{T} \), and from Lemma 14.6.1 we know that the columns of \( M{\left( {M}^{T}M\right) }^{-1} \) form an integral basis for \( {\mathcal{L}}^{ * } \) . Therefore, it is sufficient to show that if \( y \in {\mathbb{Z}}^{r} \), then \( y = {... | Yes |
Lemma 14.7.1 The flow lattice of a graph \( X \) is even if and only if \( X \) is bipartite. The cut lattice of \( X \) is even if and only if \( X \) is even. | Proof. If \( x \) and \( y \) are even vectors, then\n\n\[ \langle x + y, x + y\rangle = \langle x, x\rangle + 2\langle x, y\rangle + \langle y, y\rangle \]\n\nand so \( x + y \) is also even. If \( X \) is bipartite, then all cycles in it have even length. It follows that the flow lattice of integer flows is spanned b... | Yes |
Theorem 14.7.2 The determinant of the cut lattice of a connected graph \( X \) is equal to the number of spanning trees of \( X \) . | Proof. Let \( D \) be the oriented incidence matrix of \( X \), let \( u \) be a vertex of \( X \), and let \( {D}_{u} \) be the matrix obtained by deleting the row corresponding to \( u \) from \( D \) . Then the columns of \( {D}_{u}^{T} \) form an integral basis for the lattice, and so its determinant is \( \det \le... | Yes |
Theorem 14.7.3 The determinant of the flow lattice of a connected graph \( X \) is equal to the number of spanning trees of \( X \) . | Proof. Suppose the rows of the matrix\n\n\[ M = \left( \begin{array}{ll} I & R \end{array}\right) \]\n\nform a basis for the cut space of \( X \) (the existence of such a basis is guaranteed by the spanning-tree construction of Section 14.1). Then the rows of the matrix\n\n\[ N = \left( \begin{array}{ll} - {R}^{T} & I ... | Yes |
Theorem 14.8.1 If \( X \) is a connected graph, then the matrix\n\n\[ P = \frac{1}{\tau \left( X\right) }\mathop{\sum }\limits_{T}{N}_{T} \]\n\nrepresents orthogonal projection onto the cut space of \( X \) . | Proof. To prove the result we will show that \( P \) is symmetric, that \( {Px} = x \) for any vector in the cut space of \( X \), and that \( {Px} = 0 \) for any vector in the flow space of \( X \) .\n\nNow,\n\n\[ {\left( {N}_{T}\right) }_{eg}^{2} = \mathop{\sum }\limits_{{f \in E\left( X\right) }}{\left( {N}_{T}\righ... | Yes |
Lemma 14.9.1 In an infinite chip-firing game, every vertex is fired infinitely often. | Proof. Since there are only a finite number of ways to place \( N \) chips on \( X \), some configuration, say \( s \), must reappear infinitely often. Let \( \sigma \) be the sequence of vertices fired between two occurrences of \( s \) . If there is a vertex \( v \) that is not fired in this sequence, then the neighb... | No |
Theorem 14.9.2 Let \( X \) be a graph with \( n \) vertices and \( m \) edges and consider the chip-firing games on \( X \) with \( N \) chips. Then\n\n(a) If \( N > {2m} - n \), the game is infinite.\n\n(b) If \( m \leq N \leq {2m} - n \), the game may be finite or infinite.\n\n(c) If \( N < m \), the game is finite. | Proof. Let \( d\left( v\right) \) be the valency of the vertex \( v \) . If each vertex has at most \( d\left( v\right) - 1 \) chips on it, then\n\n\[ N \leq \mathop{\sum }\limits_{v}\left( {d\left( v\right) - 1}\right) = {2m} - n.\]\n\nSo, if \( N > {2m} - n \), there is always a vertex with as least as many chips on ... | Yes |
Theorem 14.9.3 Let \( X \) be a connected graph and let \( \sigma \) and \( \tau \) be two firing sequences starting from the same state \( s \) with respective scores \( x \) and \( y \) . Then \( \tau \) followed by \( \sigma \smallsetminus \tau \) is a firing sequence starting from \( s \) having score \( x \vee y \... | Proof. We leave the proof of this result as a useful exercise. | No |
Corollary 14.9.4 Let \( X \) be a connected graph, and \( s \) a given initial state. Then either every chip-firing game starting from \( s \) is infinite, or all such games terminate in the same state. | Proof. Let \( \tau \) be the firing sequence of a terminating game starting from \( s \), and let \( \sigma \) be the firing sequence of another game starting from \( s \) . Then by Theorem 14.9.3, \( \sigma \smallsetminus \tau \) is necessarily empty, and hence \( \sigma \) is finite.\n\nNow, suppose that \( \sigma \)... | Yes |
Lemma 14.10.1 Suppose \( u \) and \( v \) are adjacent vertices in the graph \( X \) . At any stage of a chip-firing game on \( X \) with \( N \) chips, the difference between the number of times that \( u \) has been fired and the number of times that \( v \) has been fired is at most \( N \) . | Proof. Suppose that \( u \) has been fired \( a \) times and \( v \) has been fired \( b \) times, and assume without loss of generality that \( a < b \) . Let \( H \) be the subgraph of \( X \) induced by the vertices that have been fired at most \( a \) times. Consider the number of chips currently on the subgraph \(... | Yes |
Theorem 14.10.2 If \( X \) is a connected graph with \( n \) vertices, e edges, and diameter \( D \), then a terminating chip-firing game on \( X \) ends within \( 2\mathrm{{ne}}D \) moves. | Proof. If every vertex is fired during a game, then the game is infinite, and so in a terminating game there is at least one vertex \( v \) that is never fired. By Lemma 14.10.1, a vertex at distance \( d \) from \( v \) has fired at most \( {dN} \) times, and so the total number of moves is at most \( {nDN} \) . By Th... | Yes |
Lemma 14.10.3 Let \( M \) be a positive semidefinite matrix, with largest eigenvalue \( \rho \) . Then, for all vectors \( y \) and \( z \) ,\n\n\[ \left| {{y}^{T}{Mz}}\right| \leq \rho \parallel y\parallel \parallel z\parallel \] | Proof. Since \( M \) is positive semidefinite, for any real number \( t \) we have\n\n\[ {\left( y + tz\right) }^{T}M\left( {y + {tz}}\right) \geq 0. \]\n\nThe left side here is a quadratic polynomial in \( t \), and the inequality implies that its discriminant is less than or equal to zero. This yields the following e... | Yes |
Theorem 14.10.4 Let \( X \) be a connected graph with \( n \) vertices and let \( Q \) be the Laplacian of \( X \) . If \( {Qx} = y \) and \( {x}_{n} = 0 \), then\n\n\[ \left| {{\mathbf{1}}^{T}x}\right| \leq \frac{n}{{\lambda }_{2}}\parallel y\parallel \] | Proof. Since \( Q \) is a symmetric matrix, the results of Section 8.12 show that \( Q \) has spectral decomposition\n\n\[ Q = \mathop{\sum }\limits_{{\theta \in \operatorname{ev}\left( Q\right) }}\theta {E}_{\theta } \]\n\nSince \( X \) is connected, \( \ker Q \) is spanned by \( \mathbf{1} \), and therefore\n\n\[ {E}... | Yes |
Theorem 14.11.1 A state is recurrent if and only if it is diffuse. | Proof. Suppose that \( s \) is a recurrent state, and that \( \sigma \) is a firing sequence leading from \( s \) back to itself. Let \( Y \subseteq X \) be an induced subgraph of \( X \) , and let \( v \) be the vertex of \( Y \) that first finishes firing. Then every neighbour of \( v \) in \( Y \) is fired at least ... | Yes |
Theorem 14.11.2 Let \( X \) be a connected graph with \( m \) edges. Then there is a one-to-one correspondence between diffuse states with \( m \) chips and acyclic orientations of \( X \) . | Proof. Let \( s \) be a state given, as in the proof of Theorem 14.9.2, by an acyclic orientation of \( X \) . If \( Y \) is an induced subgraph of \( X \), then the restriction of the acyclic orientation of \( X \) to \( Y \) is an acyclic orientation of \( Y \) . Hence there is some vertex whose out-valency in \( Y \... | Yes |
Lemma 14.12.2 Let \( X \) be a connected graph with \( m \) edges, and let \( t \) be a \( q \) -critical state. Then there is a \( q \) -critical state \( s \) with \( m \) chips such that \( {s}_{v} \leq {t}_{v} \) for every vertex \( v \) . | Proof. The state \( t \) is recurrent if and only if there is a permutation \( \sigma \) of \( V\left( X\right) \) that is a legal firing sequence from \( t \) . Suppose that during this firing sequence, \( v \) is the first vertex with more than \( d\left( v\right) \) chips on it when fired. Then the state obtained fr... | Yes |
Lemma 14.12.2 Let \( X \) be a connected graph with \( m \) edges, and let \( t \) be a \( q \) -critical state. Then there is a \( q \) -critical state \( s \) with \( m \) chips such that \( {s}_{v} \leq {t}_{v} \) for every vertex \( v \) . | Proof. The state \( t \) is recurrent if and only if there is a permutation \( \sigma \) of \( V\left( X\right) \) that is a legal firing sequence from \( t \) . Suppose that during this firing sequence, \( v \) is the first vertex with more than \( d\left( v\right) \) chips on it when fired. Then the state obtained fr... | Yes |
Lemma 14.13.1 In the dollar game, after \( q \) has been fired, no other vertex can be fired twice before \( q \) is fired again. | Proof. Suppose that no vertex has yet been fired twice after \( q \) and consider the number of chips on any vertex \( u \) that has been fired exactly once since \( q \) . Immediately before \( q \) was last fired, \( u \) had at most \( d\left( u\right) - 1 \) chips on it. Since then, \( u \) has gained at most \( d\... | Yes |
Lemma 14.13.2 If \( s \) and \( t \) are \( q \) -critical states such that \( s - t = {Qx} \) for some integer vector \( x \), then \( s = t \) . | Proof. We shall show that \( x \) is necessarily a constant vector, so \( {Qx} = 0 \) , and hence \( s = t \) . Assume for a contradiction that \( x \) is not constant. Then, exchanging \( s \) and \( t \) if necessary, we may assume that \( {x}_{q} \) is not a maximum coordinate of \( x \) . Let the permutation \( \ta... | Yes |
Theorem 14.13.3 Let \( X \) be a connected graph on \( n \) vertices. Each coset of \( \mathcal{L}\left( Q\right) \) in \( {\mathbb{Z}}^{n} \cap {\mathbf{1}}^{ \bot } \) contains a unique q-critical state for the dollar game. | Proof. Given a coset of \( \mathcal{L}\left( Q\right) \), choose an element \( s \) in the coset that represents a valid initial state for the dollar game. By the discussion above, every game with initial state \( s \) eventually falls into a loop containing a unique \( q \) -critical state. Therefore, each coset of \(... | Yes |
Lemma 14.14.1 Let \( a \) and \( b \) be elements of the lattice \( \mathcal{L} \) with \( \langle a, b\rangle \geq 0 \) . Then \( H\left( a\right) \cap H\left( b\right) \subseteq H\left( {a + b}\right) \) . | Proof. Suppose \( x \in H\left( a\right) \cap H\left( b\right) \) . Then\n\n\[ \langle x, a + b\rangle = \langle x, a\rangle + \langle x, b\rangle \leq \frac{1}{2}\langle a, a\rangle + \frac{1}{2}\langle b, b\rangle .\n\]\n\nSince \( \langle a, b\rangle \geq 0 \), we have that\n\n\[ \langle a + b, a + b\rangle \geq \la... | Yes |
Lemma 14.14.2 An element a of \( \mathcal{L} \) is indecomposable if and only if a and -a are the two elements of minimum norm in the coset \( a + 2\mathcal{L} \) . | Proof. Suppose \( a \in \mathcal{L} \) . If \( x \in \mathcal{L} \), then \( a = a - x + x \), whence we see that \( a \) is indecomposable if and only if\n\n\[ \langle a - x, x\rangle < 0 \]\n\nfor all elements of \( \mathcal{L} \smallsetminus \{ 0, a\} \) . Since\n\n\[ \langle a - {2x}, a - {2x}\rangle = \langle a, a... | Yes |
Theorem 14.14.3 Let \( \mathcal{V} \) be the Voronoi cell of the origin in the lattice \( \mathcal{L} \) . Then \( \mathcal{V} \) is the intersection of the closed half-spaces \( H\left( a\right) \), where a ranges over the indecomposable elements of \( \mathcal{L} \) . For each such a, the intersection \( \mathcal{V} ... | Proof. We must show that \( \mathcal{V} \cap H\left( a\right) \) has dimension one less than the dimension of the polytope. So let \( a \) be a fixed indecomposable element of \( \mathcal{L} \) and let \( u \) be any vector orthogonal to \( a \) . If \( b \) is a second indecomposable element of \( \mathcal{L} \), then... | Yes |
Theorem 14.14.4 The indecomposable vectors in the flow lattice of a connected graph \( X \) are the signed characteristic vectors of the cycles. | Proof. Let \( \mathcal{L} \) be a lattice contained in \( {\mathbb{Z}}^{n} \), and let \( x \) be an element of \( \mathcal{L} \) of minimal support which has all its entries in \( \{ - 1,0,1\} \) . For any element \( y \in \mathcal{L} \) we have \( {\left( x + 2y\right) }_{i} \neq 0 \) whenever \( {x}_{i} \neq 0 \), a... | No |
Lemma 14.15.1 Let \( X \) be a graph with \( n \) vertices and \( c \) components, with incidence matrix \( B \) . Then the 2-rank of \( B \) is \( n - c \) . | Proof. The argument given in Theorem 8.3.1 remains valid over \( {GF}\left( 2\right) \) . (The argument in Theorem 8.2.1 implicitly uses the fact that \( - 1 \neq 1 \), and hence fails over \( {GF}\left( 2\right) \) .) | Yes |
Lemma 14.15.2 A graph \( X \) is pedestrian if and only if each subgraph of \( X \) is the symmetric difference of an even subgraph and an edge cutset. | Proof. A subgraph of \( X \) is the symmetric difference of an even subgraph and an edge cutset if and only if its characteristic vector lies in \( C + F \) . | No |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.