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Theorem 20 If \( \mathbb{N} \) is coloured with finitely many colours then, for every \( \ell \geq 1 \) , one of the colour classes contains infinitely many translates of the same \( \ell \) -cube. | Proof. It clearly suffices to prove the following finite version of this result.\n\nThere is a function \( H : \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N} \) such that if \( N \geq H\left( {k, l}\right) \) then every \( k \) -colouring of \( \left\lbrack N\right\rbrack \) contains a monochromatic \( \ell \) -cu... | Yes |
Theorem 21 For every \( k \geq 1 \) there is an integer \( m \) such that every \( k \) -colouring of \( \left\lbrack m\right\rbrack \) contains integers \( x, y, z \) of the same colour such that\n\n\[ x + y = z. \] | Proof. We claim that \( m = {R}_{k}\left( 3\right) - 1 \) will do, where \( {R}_{k}\left( 3\right) = {R}_{k}\left( {3,\ldots ,3}\right) \) is the graphical Ramsey number for \( k \) colours and triangles, i.e., the minimal integer \( n \) such that every \( k \) -colouring of the edges of \( {K}_{n} \) contains a monoc... | Yes |
Theorem 22 Given \( p \) and \( k \), if \( n \) is large enough, then every \( k \) -colouring of \( \left\lbrack n\right\rbrack \) contains a monochromatic arithmetic progression of length \( p \) . | Null | No |
Theorem 23 For every \( p \) and \( k \), there is an integer \( n \) such that if \( A \) is an alphabet with \( p \) letters then every \( k \)-colouring \( c : {A}^{n} \rightarrow \left\lbrack k\right\rbrack \) contains a monochromatic line. | The Hales-Jewett function \( {HJ}\left( {p, k}\right) \) is defined much like the corresponding van der Waerden function: \( {HJ}\left( {p, k}\right) \) is the minimal value of \( n \) that will do in Theorem 23. | No |
Lemma 25 Given \( n \) and \( k \), if \( m \geq S\left( {n, k}\right) \) then the following assertions hold. Let \( S \) and \( {S}_{0} \) be the Shelah subsets of \( {\left\lbrack m\right\rbrack }^{\left\lbrack 2n\right\rbrack } \) and let \( c : S \rightarrow \left\lbrack k\right\rbrack \) . Then there is a point \(... | Null | No |
Theorem 27 A matrix with integer entries is partition regular if and only if it satisfies the columns condition. | This beautiful theorem reduces partition regularity to a property can be checked in finite time. It is worth remarking that neither of the two implications is easy. Also, as in most Ramsey type results, by the standard compactness argument we have encountered several times, the infinite version implies the finite versi... | No |
Lemma 28 If \( \mathbb{N} \) rejects \( \varnothing \) then there exists an \( M \in {\mathbb{N}}^{\left( \omega \right) } \) that rejects every \( X \subset M \) . | Proof. Note first that there is an \( {M}_{0} \) such that every \( X \subset {M}_{0} \) is either accepted or rejected by \( {M}_{0} \) . Indeed, put \( {N}_{0} = \mathbb{N},{a}_{0} = 1 \) . Suppose that we have defined \( {N}_{0} \supset {N}_{1} \supset \cdots \supset {N}_{k} \) and \( {a}_{i} \in {N}_{i} - {N}_{i + ... | Yes |
Theorem 29 Every open subset of \( {2}^{\mathbb{N}} \) is Ramsey. | Proof. Let \( \mathcal{F} \subset {2}^{\mathbb{N}} \) be open and assume that \( {A}^{\left( \omega \right) } ⊄ \mathcal{F} \) for every \( A \in {\mathbb{N}}^{\left( \omega \right) } \), i.e., \( \mathbb{N} \) rejects \( \varnothing \) . Let \( M \) be the set whose existence is guaranteed by Lemma 28. If \( {M}^{\lef... | Yes |
Corollary 30 Let \( \mathcal{G} \subset {\mathbb{N}}^{\left( < \omega \right) } \) be dense. Then there is an \( M \in {N}^{\left( \omega \right) } \) such that every \( A \subset M \) has an initial segment belonging to \( \mathcal{G} \) . | Proof. Let \( \mathcal{F} = \{ F \subset \mathbb{N} : F \) has an initial segment belonging to \( \mathcal{G}\} \) . Then \( \mathcal{F} \) is open, so there is an \( M \in {\mathbb{N}}^{\left( \omega \right) } \) such that either \( {M}^{\left( \omega \right) } \subset \mathcal{F} \), in which case we are done, or els... | Yes |
Corollary 31 Let \( \mathcal{G} \subset {\mathbb{N}}^{\left( < \omega \right) } \) be a thin family, and let \( k \in \mathbb{N} \) . Then for any \( k \) - colouring of \( \mathcal{G} \) there is an infinite set \( A \subset \mathbb{N} \) such that all members of \( \mathcal{G} \) contained in \( A \) have the same co... | Proof. It clearly suffices to prove the result for \( k = 2 \) . Consider a red and blue colouring of \( \mathcal{G} : \mathcal{G} = {\mathcal{F}}_{\text{red }} \cup {\mathcal{F}}_{\text{blue }} \) . If \( {F}_{\text{red }} \) is dense then let \( M \) be the set guaranteed by Corollary 30 . For every \( F \in \mathcal... | Yes |
Theorem 1 Let \( {X}_{s} = {X}_{s}\left( G\right) \) be the number of complete subgraphs of order \( s \) in \( G \), and let \( {X}_{s}^{\prime } = {X}_{s}^{\prime }\left( G\right) = {X}_{s}\left( \bar{G}\right) \) . Then\n\n\[ \n{\mathbb{E}}_{M}\left( {X}_{s}\right) = \left( \begin{array}{l} n \\ s \end{array}\right)... | Null | No |
Theorem 2 (i) If \( 3 \leq s \leq n \) are such that\n\n\[ \left( \begin{array}{l} n \\ s \end{array}\right) < {2}^{\left( \begin{array}{l} s \\ 2 \end{array}\right) - 1} \]\n\nthen \( R\left( {s, s}\right) \geq n + 1 \) . Also,\n\n\[ R\left( {s, s}\right) > \frac{1}{e\sqrt{2}}s{2}^{s/2}. \]\n\n(7) | Proof. (i) Consider \( \mathcal{G}\left( {n,1/2}\right) \) . With the notation above,\n\n\[ {\mathbb{E}}_{1/2}\left( {{X}_{s} + {X}_{s}^{\prime }}\right) = 2\left( \begin{array}{l} n \\ s \end{array}\right) {2}^{-\left( \begin{array}{l} s \\ 2 \end{array}\right) } < 1 \]\n\nso there is a graph \( G \in \mathcal{G}\left... | Yes |
Theorem 3 Let \( 2 \leq s \leq {n}_{1},2 \leq t \leq {n}_{2},\alpha = \left( {s - 1}\right) /\left( {{st} - 1}\right) \) and \( \beta = \) \( \left( {t - 1}\right) /\left( {{st} - 1}\right) \) . Then there is a bipartite graph \( {G}_{2}\left( {{n}_{1},{n}_{2}}\right) \) of size\n\n\[ \n\left\lfloor {\left( {1 - \frac{... | Proof. Let\n\n\[ \nn = {n}_{1} + {n}_{2} \n\]\n\n\[ \n{V}_{1} = \left\{ {1,2,\ldots ,{n}_{1}}\right\} \n\]\n\n\[ \n{V}_{2} = \left\{ {{n}_{1} + 1,{n}_{1} + 2,\ldots ,{n}_{1} + {n}_{2}}\right\} \n\]\n\n\[ \nE = \left\{ {{ij} : i \in {V}_{1}, j \in {V}_{2}}\right\} \n\]\n\n\[ \nM = \left\lfloor {{n}_{1}^{1 - \alpha }{n}_... | Yes |
Theorem 5 Let \( 1 \leq h \leq k \) be fixed natural numbers and let \( 0 < p < 1 \) be fixed also. Then in \( \mathcal{G}\left( {n, p}\right) \) a.e. graph \( {G}_{p} \) is such that for every sequence of \( k \) vertices \( {x}_{1},{x}_{2},\ldots ,{x}_{k} \) there exists a vertex \( x \) such that \( x{x}_{i} \in E\l... | Proof. Let \( {x}_{1},{x}_{2},\ldots ,{x}_{k} \) be a sequence of vertices. The probability that a vertex \( x \in W = V\left( G\right) - \left\{ {{x}_{1},\ldots ,{x}_{k}}\right\} \) has the required properties is \( {p}^{h}{q}^{k - h} \) . Since for \( x, y \in W, x \neq y \), the edges \( x{x}_{i} \) are chosen indep... | Yes |
Theorem 6 Let \( 0 < p = p\left( n\right) < 1 \) be such that \( p{n}^{2} \rightarrow \infty \) and \( \left( {1 - p}\right) {n}^{2} \rightarrow \infty \) as \( n \rightarrow \infty \), and let \( Q \) be a property of graphs.\n\n(i) Suppose \( \varepsilon > 0 \) is fixed and, if \( \left( {1 - \varepsilon }\right) N <... | Null | No |
Theorem 7 Let \( k \geq 2, k - 1 \leq \ell \leq \left( \begin{aligned} k \\ 2 \end{aligned}\right) \) and let \( F = G\left( {k, l}\right) \) be a balanced graph (with \( k \) vertices and \( \ell \) edges). If \( p\left( n\right) {n}^{k/l} \rightarrow 0 \) then almost no \( {G}_{n, p} \) contains \( F \) , and if \( p... | Proof. Let \( p = \gamma {n}^{-k/\ell },0 < \gamma < {n}^{k/\ell } \), and denote by \( X = X\left( G\right) \) the number of copies of \( F \) contained in \( {G}_{n, p} \) . Denote by \( {k}_{F} \) the number of graphs with a fixed set of \( k \) labelled vertices that are isomorphic to \( F \) . Clearly, \( {k}_{F} ... | Yes |
Theorem 10 Let \( 0 < p < 1 \) be constant. Then\n\n\[ \chi \left( {G}_{n, p}\right) = \left( {\frac{1}{2} + o\left( 1\right) }\right) \frac{\log n}{\log \left( {1/q}\right) } \]\n\nfor a.e. \( {G}_{n, p} \), where \( q = 1 - p \) . | What Theorem 10 claims is that if \( \varepsilon > 0 \) then\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}{\mathbb{P}}_{p}\left( {\left| {\chi \left( {G}_{n, p}\right) \log \left( {1/q}\right) /\log n - \frac{1}{2}}\right| < \varepsilon }\right) = 1. \] | No |
Theorem 11 For almost every graph process \( \widetilde{G} \) we have \( \tau \left( {\widetilde{G};\operatorname{conn}}\right) = \tau (\widetilde{G} \) ; \( \delta \geq 1) \) . | Null | No |
Lemma 12 Let \( c > 3 \) and \( 0 < \gamma < \frac{1}{3} \) be constants and let \( p = \left( {c\log n}\right) /n \) . Then in \( \mathcal{G}\left( {n, p}\right) \) we have\n\n\[ \n{\mathbb{P}}_{p}\left( {{D}_{t} > 0\text{ for some }t,1 \leq t \leq {\gamma n}}\right) = O\left( {n}^{3 - c}\right) .\n\] | Proof. Put \( \beta = \frac{\left( c - 3\right) }{4c} \) . Clearly,\n\n\[ \n\mathop{\sum }\limits_{{t = 1}}^{{\lfloor {\gamma n}\rfloor }}{\mathbb{E}}_{p}\left( {D}_{t}\right) = \mathop{\sum }\limits_{{t = 1}}^{{\lfloor {\gamma n}\rfloor }}\left( \begin{array}{l} n \\ t \end{array}\right) \left( \begin{matrix} n - t \\... | Yes |
Theorem 14 Let \( \omega \left( n\right) \rightarrow \infty \) and set \( {p}_{\ell } = \left( {\log n + \log \log n - \omega \left( n\right) }\right) /n \) and \( {p}_{u} = \left( {\log n + \log \log n + \omega \left( n\right) }\right) /n \) . Then \( {p}_{\ell } \) is a lower threshold function for the property of be... | Null | No |
Theorem 16 Almost every random graph process is such that if \( k \geq 2 \) is fixed and \( t = o\left( {n}^{\left( {k - 1}\right) /k}\right) \) then every component of \( {G}_{t} \) is a tree of order at most \( t \) . Furthermore, if \( s \) is constant and \( t/{n}^{\left( {k - 2}\right) /\left( {k - 1}\right) } \ri... | The proof of this assertion goes along the lines of the proof of Theorem 7 and is rather vapid: we do not even need that there are \( {k}^{k - 2} \) trees of order \( k \) (see Exercise I. 41 and Theorem VIII.20). All we have to do is to estimate \( \mathbb{E}\left( {X}_{k}\right) \) and \( \mathbb{E}\left( {X}_{k}^{2}... | No |
Theorem 17 Let \( c > 0 \) and \( h \geq 1 \) be fixed and let \( \omega \left( n\right) \rightarrow \infty \) . Set \( \alpha = \) \( c - 1 - \log c \) and \( t = t\left( n\right) = \lfloor {cn}/2\rfloor \) .\n\n(i) If \( c < 1 \) then, for almost every random graph \( {G}_{t} \) ,\n\n\[ \left| {{L}^{\left( i\right) }... | Null | No |
Theorem 18 Let \( a \geq 2 \) be fixed. If \( n \) is sufficiently large, \( \varepsilon = \varepsilon \left( n\right) < 1/3 \) and \( p = p\left( n\right) = \frac{1 + \varepsilon }{n} \) then, with probability at least \( 1 - {n}^{-a},{G}_{n, p} \) has no component whose order \( k \) satisfies\n\n\[ \frac{8a}{{\varep... | Proof. Set \( {k}_{0} = \left\lceil {{8a}{\varepsilon }^{-2}\log n}\right\rceil \) and \( {k}_{1} = \left\lceil {{\varepsilon }^{2}n/{12}}\right\rceil \) . Writing \( {p}_{k} \) for the probability that the component of \( {G}_{n, p} \) containing a fixed vertex has \( k \) vertices, the probability that \( {G}_{n, p} ... | Yes |
Theorem 19 Let \( {\left( {Z}_{n}\right) }_{0}^{\infty } \) be as above, with \( c > 1 \), and write \( {p}_{\infty } \) for the probability that \( {Z}_{n} > 0 \) for every \( n \) . Then \( {p}_{\infty } \) is the unique root of\n\n\[ \n{e}^{-c{p}_{\infty }} = 1 - {p}_{\infty } \n\]\n\nin the interval \( \left( {0,1}... | Proof. Let \( {p}_{n} \) be the probability that \( {Z}_{n} > 0 \), so that \( {p}_{0} = 1 \) and \( {p}_{\infty } = \) \( \mathop{\lim }\limits_{{n \rightarrow \infty }}{p}_{n} \) . First we check, by induction on \( n \), that \( {p}_{n} \geq \gamma \) for every \( n \), where \( \gamma \) is the unique root of \( {e... | Yes |
Theorem 1 If \( \left( {1/p}\right) + \left( {1/q}\right) + \left( {1/r}\right) > 1 \) then the group \( \left( {p, q, r}\right) \) is finite and has order \( {2s} \) where \( 1/s = \left( {1/p}\right) + \left( {1/q}\right) + \left( {1/r}\right) - 1 \) . The Cayley diagram is a tessellation of the sphere (as in the fir... | Null | No |
Theorem 2 The subgroup \( B \) of \( A \) is generated by the decorations of the chords. | In particular, the subgroup \( B \) in Fig. VIII. 9 is generated by \( b, a{b}^{3}{a}^{-1} \) , \( a{b}^{-1}a{b}^{-1}{a}^{-1} \) and \( {abab}{a}^{-1} \) . | Yes |
Theorem 4 A subgroup of a free group is free. Furthermore, if \( A \) is a free group of rank \( k \) (that is, it has \( k \) free generators) and \( B \) is a subgroup of index \( n \), then \( B \) has rank \( \left( {k - 1}\right) n + 1 \) . | Proof. The presentation of \( B \) given in Theorem 3 is a free presentation on the set of chords of the Schreier diagram. Altogether there are \( {kn} \) edges of which \( n - 1 \) are tree edges; hence there are \( \left( {k - 1}\right) n + 1 \) chords. | Yes |
Theorem 4 A subgroup of a free group is free. Furthermore, if \( A \) is a free group of rank \( k \) (that is, it has \( k \) free generators) and \( B \) is a subgroup of index \( n \), then \( B \) has rank \( \left( {k - 1}\right) n + 1 \) . | Proof. The presentation of \( B \) given in Theorem 3 is a free presentation on the set of chords of the Schreier diagram. Altogether there are \( {kn} \) edges of which \( n - 1 \) are tree edges; hence there are \( \left( {k - 1}\right) n + 1 \) chords. | Yes |
Theorem 5 Let \( G \) be a connected graph of order \( n \) with adjacency matrix \( A \) . (i) Every eigenvalue \( \mu \) of \( G \) satisfies \( \left| \mu \right| \leq \Delta = \Delta \left( G\right) \) . | Proof. (i) Let \( \mathbf{x} = \left( {x}_{i}\right) \) be a non-zero eigenvector with eigenvalue \( \mu \) . Let \( {x}_{p} \) be a weight with maximum modulus: \( \left| {x}_{p}\right| \geq \left| {x}_{i}\right| \) for every \( i \) ; we may assume without loss of generality that \( {x}_{p} = 1 \) . Then \[ \left| \m... | Yes |
Corollary 6 Every graph \( G \) satisfies \( \chi \left( G\right) \leq {\mu }_{\max }\left( G\right) + 1 \) . | Proof. For every induced subgraph \( H \) of \( G \) we have\n\n\[ \delta \left( H\right) \leq {\mu }_{\max }\left( H\right) \leq {\mu }_{\max }\left( G\right) \]\n\nso we are done by Theorem V.1. | Yes |
Theorem 7 Let \( G \) be a non-empty graph. Then\n\n\[ \chi \left( G\right) \geq 1 - \frac{{\mu }_{\max }\left( G\right) }{{\mu }_{\min }\left( G\right) }.\] | Proof. As before, we take \( V = \left\{ {{v}_{1},\ldots ,{v}_{n}}\right\} \) for the set of vertices, so that \( \left( {{v}_{1},\ldots ,{v}_{n}}\right) \) is the canonical basis of \( {C}_{0}\left( G\right) \) . Let \( c : V\left( G\right) \rightarrow \left\lbrack k\right\rbrack \) be a (proper) colouring of \( G \) ... | Yes |
Theorem 8 The adjacency matrix of a graph \( G \) has at least \( \beta \left( G\right) \) non-negative and at least \( \beta \left( G\right) \) non-positive eigenvalues, counted with multiplicity. | Proof. The Lagrangian \( {f}_{G}\left( \mathbf{x}\right) \) is identically 0 on every subspace spanned by a set of independent vertices. In particular, \( {f}_{G} \) is positive semi-definite and negative semidefinite on a subspace of dimension \( \beta \left( G\right) \) . Hence we are done by the analogues of (1). | Yes |
Theorem 9 The complete graph \( {K}_{n} \) is not the edge-disjoint union of \( n - 2 \) complete bipartite graphs. | Proof. Suppose that, contrary to the assertion, \( {K}_{n} \) is the edge-disjoint union of complete bipartite graphs \( {G}_{1},\ldots ,{G}_{n - 2} \) . For each \( i \), let \( {H}_{i} \) be obtained from \( {G}_{i} \) by adding to it isolated vertices so that \( V\left( {H}_{i}\right) = V\left( {K}_{n}\right) \) . N... | Yes |
Theorem 10 Let \( G \) be a graph with clique number \( {k}_{0} \) . Then \( f\left( G\right) = \left( {{k}_{0} - 1}\right) /{k}_{0} \) . | Proof. Let \( \mathbf{y} = {\left( {y}_{i}\right) }_{1}^{n} \in S \) be a point at which \( {f}_{G}\left( \mathbf{x}\right) \) attains its maximum and \( \operatorname{supp}\mathbf{y} = \left\{ {{v}_{i} : {y}_{i} > 0}\right\} \) is as small as possible. We claim that the support of \( \mathbf{y} \) spans a complete sub... | Yes |
Corollary 11 Let \( G = G\left( {n, m}\right) \), with \( m > \frac{r - 2}{2\left( {r - 1}\right) }{n}^{2} \) . Then \( G \) contains a complete graph of order \( r \) . | Proof. Writing \( {k}_{0} = \omega \left( G\right) \) for the clique number of \( G \), we know that \( f\left( G\right) = \) \( \left( {{k}_{0} - 1}\right) /{k}_{0} \) . On the other hand, with \( \mathbf{x} = \left( {1/n,1/n,\ldots ,1/n}\right) \) we see that\n\n\[ f\left( G\right) \geq {f}_{G}\left( \mathbf{x}\right... | Yes |
Theorem 12 The vertex connectivity of an incomplete graph \( G \) is at least as large as the second smallest eigenvalue \( {\lambda }_{2}\left( G\right) \) of the Laplacian of \( G \) . | Proof. If \( G = {K}_{n} \) then \( {\lambda }_{2} = n - 1 = \kappa \left( G\right) \) . Suppose then that \( G \) is not a complete graph, and let \( {V}^{\prime } \cup S \cup {V}^{\prime \prime } \) be a partition of the vertex set \( \left\{ {{v}_{1},\ldots ,{v}_{n}}\right\} \) of \( G \) such that \( \left| S\right... | Yes |
Theorem 13 Let \( G \) be a graph of order \( n \) . Then for \( U \subset V = V\left( G\right) \) we have\n\n\[ \left| {\partial U}\right| \geq \frac{{\lambda }_{2}\left( G\right) \left| U\right| \left| {V \smallsetminus U}\right| }{n}. \] | Proof. We may assume that \( \varnothing \neq U \neq V = \left\{ {{v}_{1},\ldots ,{v}_{n}}\right\} \) . Set \( k = \left| U\right| \), and define \( \mathbf{x} = \mathop{\sum }\limits_{{i = 1}}^{n}{x}_{i}{v}_{i} \) as follows:\n\n\[ {x}_{i} = \left\{ \begin{array}{ll} n - k & \text{ if }{v}_{i} \in U, \\ - k & \text{ i... | Yes |
Theorem 14 Let \( G \) be a connected \( k \) -regular graph of order \( n \), with adjacency matrix \( A \) and distinct eigenvalues \( k,{\mu }_{1},{\mu }_{2},\ldots ,{\mu }_{r} \) . Then\n\n\[ \mathop{\prod }\limits_{{i = 1}}^{r}\frac{A - {\mu }_{i}I}{k - {\mu }_{i}} = \frac{J}{n} \]\n | Proof. Each side is the orthogonal projection onto \( \langle \mathbf{j}\rangle \) . | Yes |
Theorem 15 Let \( G, A, C, P \) and \( {\pi }_{r} \) be as above.\n\n(i) \( {\pi }_{r}A = C{\pi }_{r} \), that is, the diagram below commutes.\n\n\n\n(ii) The adjacency matrix \( A \) and the collapsed adjacency matr... | Proof. (i) Let us show that \( {\pi }_{r}\left( {A{v}_{t}}\right) = C\left( {{\pi }_{r}{v}_{t}}\right) \), where \( {v}_{t} \) is the basis vector corresponding to an arbitrary vertex \( {v}_{t} \in {V}_{i}^{\left( r\right) } \) . To do this it suffices to check that the \( i \) th coordinates of the two sides are equa... | Yes |
Theorem 16 Let \( G \) be a connected highly regular graph of order \( n \) with collapsed adjacency matrix \( C \) . Let \( {\mu }_{1},{\mu }_{2},\ldots ,{\mu }_{r} \) be the roots of the characteristic polynomial of \( C \) different from \( k \), the degree of the vertices of \( G \) . Then there are natural numbers... | Proof. We know from Theorem 5 that \( {\mu }_{1},{\mu }_{2},\ldots ,{\mu }_{r} \) are the eigenvalues of \( A \) in addition to \( k \), which has multiplicity 1 . Thus if \( m\left( {\mu }_{i}\right) \) is the multiplicity of \( {\mu }_{1} \) then\n\n\[ 1 + \mathop{\sum }\limits_{{i = 1}}^{r}m\left( {\mu }_{i}\right) ... | Yes |
Theorem 17 Let \( G \) be a connected imcomplete regular graph. Then \( G \) is strongly regular iff it has precisely three distinct eigenvalues. | Proof. Suppose \( G \) is a strongly regular graph with adjacency matrix \( A \) . As its collapsed adjacency matrix has order 3 , by Theorem 15 it has at most three distinct eigenvalues. Furthermore, if \( G \) had only two distinct eigenvalues then, by Theorem 14 we would have \( A \in \langle I, J\rangle \), which w... | Yes |
Theorem 18 If there is a strongly regular graph of order \( n \) with parameters \( \left( {k, a, b}\right) \) then\n\n\[ \n{m}_{1},{m}_{2} = \frac{1}{2}\left\{ {n - 1 \pm \frac{\left( {n - 1}\right) \left( {b - a}\right) - {2k}}{{\left\{ {\left( a - b\right) }^{2} + 4\left( k - b\right) \right\} }^{1/2}}}\right\} \n\]... | Proof. The characteristic polynomial of the collapsed adjacency matrix \( C \) is\n\n\[ \n{x}^{3} + \left( {b - a - k}\right) {x}^{2} + \left( {\left( {a - b}\right) k + b - k}\right) x + k\left( {k - b}\right) .\n\]\n\nOn dividing by \( x - k \), we find that the roots different from \( k \) are\n\n\[ \n{\mu }_{1},{\m... | Yes |
Theorem 19 Suppose there is a \( k \) -regular graph \( G \) of order \( n = {k}^{2} + 1 \) and diameter 2. Then \( k = 2,3,7 \) or 57 . | Proof. We know from Theorem IV. 1 that \( G \) is strongly regular with parameters \( \left( {k,0,1}\right) \) . By the rationality condition at least one of the following two conditions has to hold:\n\n(i): \( \left( {n - 1}\right) - {2k} = {k}^{2} - {2k} = 0 \) and \( n - 1 = {k}^{2} \) is even,\n\n(ii): \( 1 + 4\lef... | Yes |
Corollary 21 Let \( {d}_{1} \leq {d}_{2} \leq \cdots \leq {d}_{n} \) be the degree sequence of a tree: \( {d}_{1} \geq 1 \) and \( \mathop{\sum }\limits_{{i = 1}}^{n}{d}_{i} = {2n} - 2 \) . Then the number of labelled trees of order \( n \) with degree sequence \( {\left( {d}_{i}\right) }_{1}^{n} \) is given by the mul... | \[ \left( \begin{matrix} n - 2 \\ {d}_{1} - 1,{d}_{2} - 1,\ldots ,{d}_{n} - 1 \end{matrix}\right) . \] | Yes |
Lemma 22 \( \left| \Gamma \right| \mathop{\sum }\limits_{{i = 1}}^{\ell }w\left( {O}_{i}\right) = \mathop{\sum }\limits_{{\alpha \in \Gamma }}\mathop{\sum }\limits_{{x \in F\left( \alpha \right) }}w\left( x\right) \) . | Proof.\n\n\[ \mathop{\sum }\limits_{{\alpha \in \Gamma }}\mathop{\sum }\limits_{{x \in F\left( \alpha \right) }}w\left( x\right) = \mathop{\sum }\limits_{{x \in X}}\mathop{\sum }\limits_{{\alpha \in \Gamma \left( x\right) }}w\left( x\right) = \mathop{\sum }\limits_{{i = 1}}^{\ell }\mathop{\sum }\limits_{{x \in {O}_{i}}... | Yes |
Theorem 23 With the notation above,\n\n\\[ \n\\left| \\Gamma \\right| S = \\widetilde{Z}\\left( {\\Gamma ;{s}_{1},{s}_{2},\\ldots ,{s}_{d}}\\right) \n\\] | Proof. By Lemma 22,\n\n\\[ \n\\left| \\Gamma \\right| S = \\left| \\Gamma \\right| \\mathop{\\sum }\\limits_{{i = 1}}^{\\ell }w\\left( {O}_{i}\\right) = \\mathop{\\sum }\\limits_{{\\alpha \\in \\Gamma }}\\mathop{\\sum }\\limits_{{f \\in F\\left( {\\alpha }^{ * }\\right) }}w\\left( f\\right) .\n\\] \n\nNow, clearly \\( ... | Yes |
Theorem 1 Let \( N = \left( {G, r}\right) \) be an electrical network, \( {s}_{1},\ldots ,{s}_{k} \in V\left( G\right) \), and \( {V}_{{s}_{1}},\ldots ,{V}_{{s}_{k}} \in \mathbb{R} \) . Then there are absolute potentials \( {V}_{x}, x \in V\left( G\right) \smallsetminus \left\{ {{s}_{1},\ldots ,{s}_{k}}\right\} \) such... | Proof. Since the energy function \( E \) is a continuous function of the absolute potentials \( \left( {V}_{x}\right) \in {\mathbb{R}}^{V\left( G\right) } \), and \( E \rightarrow \infty \) as \( \max \left| {V}_{x}\right| \rightarrow \infty \), the infimum of \( E \) is indeed attained at some \( \left( {V}_{x}\right)... | Yes |
Theorem 2 Let \( N = \left( {G, r}\right) \) be an electrical network, \( {s}_{1}\ldots ,{s}_{k} \in V\left( G\right) \), and let \( {u}_{{s}_{1}},\ldots ,{u}_{{s}_{k}} \in \mathbb{R} \), with \( \mathop{\sum }\limits_{{i = 1}}^{k}{u}_{{s}_{i}} = 0 \) . Consider the energy function\n\n\[ \nE = E\left( u\right) = \matho... | Proof. Once again, compactness implies that the infimum of \( E\left( u\right) \) is attained at some flow \( u = \left( {u}_{xy}\right) \) . Given a cycle \( {x}_{1}{x}_{2}\cdots {x}_{\ell },{x}_{\ell + 1} \equiv {x}_{1} \), let \( u\left( \varepsilon \right) \) be the flow obtained from \( u \) by increasing each \( ... | Yes |
Theorem 3 Let \( u = \left( {u}_{xy}\right) \) be a flow from \( s \) to \( t \) with value\n\n\[ \n{u}_{s} = \mathop{\sum }\limits_{{y \in \Gamma \left( s\right) }}{u}_{sy} = - \mathop{\sum }\limits_{{z \in \Gamma \left( t\right) }}{u}_{tz} = - {u}_{t} \n\]\n\ni.e., let \( u \) be a flow satisfying KCL at each vertex ... | Proof. The right-hand side is\n\n\[ \n\mathop{\sum }\limits_{{x \in V\left( G\right) }}{V}_{x}\left( {\mathop{\sum }\limits_{{y \in {\Gamma }^{ + }\left( x\right) }}{u}_{xy} - \mathop{\sum }\limits_{{z \in {\Gamma }^{ - }\left( x\right) }}{u}_{zx}}\right) = {V}_{s}{u}_{s} + {V}_{t}{u}_{t} = \left( {{V}_{s} - {V}_{t}}\r... | Yes |
Corollary 4 The total energy in an electric current from \( s \) to \( t \) is \( \left( {{V}_{s} - {V}_{t}}\right) {w}_{s} \) , where \( {w}_{s} = \mathop{\sum }\limits_{{x \in \Gamma \left( s\right) }}{w}_{sx} \) is the value of the current. If \( {V}_{s} - {V}_{t} = 1 \) then the total energy is equal to the size of... | Proof. This is immediate from Theorem 3. | Yes |
Corollary 5 The effective conductance \( {C}_{\mathrm{{eff}}}\left( {s, t}\right) \) of a network between \( s \) and \( t \) is | \[ {C}_{\text{eff }}\left( {s, t}\right) = \inf \left\{ {\mathop{\sum }\limits_{{{xy} \in E\left( G\right) }}\frac{{\left( {V}_{x} - {V}_{y}\right) }^{2}}{{r}_{xy}} : {V}_{s} = 1,{V}_{t} = 0}\right\} . \] | Yes |
Corollary 6 The effective resistance \( {R}_{\text{eff }}\left( {s, t}\right) \) of a network between \( s \) and \( t \) is\n\n\[ \n{R}_{\text{eff }}\left( {s, t}\right) = \inf \left\{ {\mathop{\sum }\limits_{{{xy} \in E\left( G\right) }}{u}_{xy}^{2}{r}_{xy} : \left( {u}_{xy}\right) \text{ is an }s\text{-th flow of si... | Proof. If \( {r}_{{x}_{0}{y}_{0}} \) is increased then the expression for \( {C}_{\text{eff }}\left( {s, t}\right) \) in Corollary 5 does not increase. Equivalently, the expression for \( {R}_{\text{eff }}\left( {s, t}\right) \) in Corollary 6 does not decrease. | No |
Corollary 7 If the resistance of a wire is increased then the effective resistance (between two vertices) does not decrease. In particular, if a wire is cut, the effective resistance does not decrease, and if two vertices are shorted, the effective resistance does not increase. | Proof. If \( {r}_{{x}_{0}{y}_{0}} \) is increased then the expression for \( {C}_{\text{eff }}\left( {s, t}\right) \) in Corollary 5 does not increase. Equivalently, the expression for \( {R}_{\text{eff }}\left( {s, t}\right) \) in Corollary 6 does not decrease. | Yes |
Theorem 8 Let \( N = \left( {G, c}\right) \) be a connected electrical network, and let \( s, t \in \) \( V\left( G\right), s \neq t \) . For \( x \in V\left( G\right) \) define\n\n\[ \n{V}_{x} = \mathbb{P}\text{(starting at}x\text{, we get to}s\text{before we get to}t\text{),} \n\]\n\nso that \( {V}_{s} = 1 \) and \( ... | Proof. By considering the very first step of the RW started at \( x \neq s, t \), we see that\n\n\[ \n{V}_{x} = \mathop{\sum }\limits_{y}{P}_{xy}{V}_{y} = \mathop{\sum }\limits_{{y \in \Gamma \left( x\right) }}\frac{{c}_{xy}}{{C}_{x}}{V}_{y} \n\]\n\nso (4) follows:\n\n\[ \n{C}_{x}{V}_{x} = \mathop{\sum }\limits_{{y \in... | Yes |
Theorem 9 Let \( N = \left( {G, c}\right) \) be a connected electrical network with \( s, t \in V\left( G\right) \) , \( s \neq t \) . For \( x \in V\left( G\right) \), set \( {V}_{x} = {S}_{x}\left( {s \rightarrow t}\right) /{C}_{x} \) . Furthermore, for \( {xy} \in E\left( G\right) \) , denote by \( {E}_{xy} \) the e... | Proof. We know that \( {S}_{t} = 0 \), so \( {V}_{t} = 0 \) . Let us check that \( \left( {V}_{x}\right) \) satisfies (4) for every \( x \neq s, t \) . Indeed, we get to \( x \) from one of its neighbours, so\n\n\[ \n{S}_{x} = \mathop{\sum }\limits_{{y \in \Gamma \left( x\right) }}{S}_{y}{P}_{yx} = \mathop{\sum }\limit... | Yes |
Theorem 10 The RW on a connected, locally finite, infinite electrical network is transient iff the effective resistance between a vertex \( s \) and \( \infty \) is finite, and it is recurrent iff the effective resistance is infinite. | Although it is intuitively clear what Theorem 10 means and how it follows from Theorem 8, let us be a little more pedantic.\n\nLet us fix a vertex \( s \) and, for \( l \in \mathbb{N} \), let \( {N}_{l} \) be the network obtained from \( N \) by shorting all the vertices at distance at least \( l \) from \( s \) to for... | No |
Theorem 11 The effective resistance \( {R}_{\text{eff }}^{\left( \infty \right) } \) of \( N \) between \( s \) and infinity is at most \( r \) iff there is a current \( \left( {u}_{xy}\right) \) in the network \( N \) such that a flow of size 1 enters the network at \( s \), at no other vertex does any current enter o... | Proof. Suppose that \( {R}_{\text{eff }}^{\left( l\right) } \leq r \) for every \( l \) . Corollary 6 guarantees a flow \( {u}^{\left( l\right) } \) of size 1 from \( s \) to \( {t}_{l} \) in \( {N}_{l} \), with total energy at most \( r \) . By compactness, a subsequence of \( \left( {u}^{\left( l\right) }\right) \) c... | Yes |
Theorem 12 We have \( {C}_{\text{eff }}^{\left( \infty \right) } \leq C \) iff for every \( {C}^{\prime } > C \) there is a function \( \left( {V}_{x}\right) \) on the vertex set \( V\left( G\right) \) such that \( {V}_{s} = 1,{V}_{x} = 0 \) for all but finitely many vertices \( x \) , and \( \mathop{\sum }\limits_{{{x... | Null | No |
Theorem 13 Consider the RW on a connected, locally finite infinite electrical network \( N = \left( {G, c}\right) = \left( {G,1/r}\right) \), where \( c = 1/r \) is the conductance and \( r \) is the resistance. This \( {RW} \) is transient iff there is a flow \( \left( {u}_{xy}\right) \) of finite energy \( \mathop{\s... | Null | No |
Theorem 15 We have \( \mathop{\lim }\limits_{{k \rightarrow \infty }}\mathbb{E}\left( {{S}_{k}\left( x\right) /k}\right) = d\left( x\right) /{2m} \), and \( {\left( {S}_{k}\left( x\right) /k\right) }_{x \in V} \) tends to \( \pi \) in probability as \( k \rightarrow \infty \) . | Proof. Note first that\n\n\[ \mathbb{E}\left( {{S}_{k}\left( x\right) }\right) = \mathop{\sum }\limits_{{i = 1}}^{k}\mathbb{P}\left( {{X}_{i} = x}\right) \]\n\nso\n\n\[ \mathop{\lim }\limits_{{k \rightarrow \infty }}\mathbb{E}\left( {{S}_{k}\left( x\right) /k}\right) = \mathop{\lim }\limits_{{k \rightarrow \infty }}\fr... | Yes |
Theorem 16 The mean return time to a vertex \( x \) in a connected graph is \( H\left( {x, x}\right) = {2m}/d\left( x\right) . \) | Proof. Set \( {Y}_{0} = 0 \) and let \( {Y}_{\ell } \) be the time our random walk \( {\left( {X}_{i}\right) }_{0}^{\infty } \) returns to \( x \) for the \( \ell \) th time when started at \( {X}_{0} = x \) . Then \( {Y}_{1} = {Y}_{1} - {Y}_{0},{Y}_{2} - {Y}_{1},{Y}_{3} - {Y}_{2},\ldots \) are i.i.d. random variables,... | Yes |
Theorem 17 Let \( {xy} \) be a fixed edge of our graph \( G \) . The expected time it takes for the simple random walk on \( G \), started at \( x \), to return to \( x \) through \( {yx} \) is \( {2m} \) . Thus if \( {X}_{0},{X}_{1},{X}_{2},\ldots \) is our SRW, with \( {X}_{0} = x \), and \( Z = \min \left\{ {k \geq ... | Proof. The probability that we pass through \( {yx} \) at time \( k + 1 \) is\n\n\[ \mathbb{P}\left( {{X}_{k} = y,{X}_{k + 1} = x}\right) = \frac{\mathbb{P}\left( {{X}_{k} = y}\right) }{d\left( y\right) }.\]\n\nTherefore, writing \( {S}_{k}\left( {yx}\right) \) for the number of times we pass through \( {yx} \) up to t... | Yes |
Theorem 18 Let \( G \) be a connected graph of order \( n \) and size \( m \) . The mean hitting times \( H\left( {x, y}\right) \) of the SRW on \( G \) satisfy\n\n\[ \n\frac{1}{2m}\mathop{\sum }\limits_{{x \in V\left( G\right) }}\mathop{\sum }\limits_{{y \in \Gamma \left( x\right) }}H\left( {x, y}\right) = n - 1.\n\]\... | Proof. Let \( \pi = \left( {\pi }_{x}\right) \) be the stationary distribution for the transition matrix \( P = \) \( \left( {P}_{xy}\right) \), so that \( {\pi P} = \pi \) and \( {\pi }_{x}{P}_{xy} = 1/{2m} \) for \( {xy} \in E\left( G\right) \) . Then\n\n\[ \n\frac{1}{2m}\mathop{\sum }\limits_{x}\mathop{\sum }\limits... | Yes |
Theorem 19 With the notation above,\n\n\[ \n{P}_{\mathrm{{esc}}}\left( {s \rightarrow t}\right) = \frac{{C}_{\mathrm{{eff}}}\left( {s, t}\right) }{d\left( s\right) } = \frac{H\left( {s, s}\right) }{C\left( {s, t}\right) } = \frac{2m}{d\left( s\right) C\left( {s, t}\right) }.\n\]\n\n(25)\n\nFurthermore,\n\n\[ \nC\left( ... | Proof. The first equality in (25) follows from relation (13) in Theorem 8. To see the other equalities in (25), let \( R \) be the first time the random walk returns to \( s \), and let \( A \) be the first time it returns to \( s \) after having visited \( t \) . Then \( \mathbb{E}\left( R\right) = H\left( {s, s}\righ... | Yes |
Theorem 20 For a connected graph \( G \), vertices \( s \neq t \), and edge \( {xy} \in E\left( G\right) \) we have\n\n\[ \n{R}_{\text{eff }}\left( {s, t}\right) = {S}_{xy}\left( {s \leftrightarrow t}\right) = \frac{{S}_{x}\left( {s \rightarrow t}\right) }{d\left( x\right) } + \frac{{S}_{x}\left( {t \rightarrow s}\righ... | Proof. With the notation in Theorem \( 9,{S}_{x}\left( {s \leftrightarrow t}\right) \) is\n\n\[ \n{S}_{x}\left( {s \rightarrow t}\right) + {S}_{x}\left( {t \rightarrow s}\right) = {V}_{x}\left( {s \rightarrow t}\right) d\left( x\right) + {V}_{x}\left( {t \rightarrow s}\right) d\left( x\right) .\n\]\n\nBut\n\n\[ \n{V}_{... | Yes |
Theorem 22 Let \( s, t \) and \( u \) be distinct vertices of a graph \( G \) . Then\n\n\[ d\left( s\right) {P}_{\mathrm{{esc}}}\left( {s \rightarrow t < u}\right) = d\left( t\right) {P}_{\mathrm{{esc}}}\left( {t \rightarrow s < u}\right) . | Proof. Let \( {W}_{s, t;u} \) be the set of walks \( W = {x}_{0}{x}_{1}\cdots {x}_{\ell } \) in \( G - u \) such that \( {x}_{i} = s \) iff \( i = 0 \) and \( {x}_{i} = t \) iff \( i = \ell \) . Then, writing \( {\left( {X}_{i}\right) }_{0}^{\infty } \) for our random walk,\n\n\[ {P}_{\mathrm{{esc}}}\left( {s \rightarr... | Yes |
Theorem 22 Let \( s, t \) and \( u \) be distinct vertices of a graph \( G \) . Then\n\n\[ d\left( s\right) {P}_{\mathrm{{esc}}}\left( {s \rightarrow t < u}\right) = d\left( t\right) {P}_{\mathrm{{esc}}}\left( {t \rightarrow s < u}\right) . | Proof. Let \( {W}_{s, t;u} \) be the set of walks \( W = {x}_{0}{x}_{1}\cdots {x}_{\ell } \) in \( G - u \) such that \( {x}_{i} = s \) iff \( i = 0 \) and \( {x}_{i} = t \) iff \( i = \ell \) . Then, writing \( {\left( {X}_{i}\right) }_{0}^{\infty } \) for our random walk,\n\n\[ {P}_{\mathrm{{esc}}}\left( {s \rightarr... | Yes |
Theorem 23 Let \( s \) , \( t \) and \( u \) be vertices of a graph \( G \) . Then\n\n\[ H\left( {s, t}\right) + H\left( {t, u}\right) + H\left( {u, s}\right) = H\left( {s, u}\right) + H\left( {u, t}\right) + H\left( {t, s}\right) .\n\] | Proof. The left-hand side is the expected time it takes to go from \( s \) to \( t \), then on to \( u \) and, finally, back to \( s \), and the right-hand side is the expected length of a tour in the opposite direction. Thus, writing \( \tau \) for the first time a walk starting at \( s \) completes a tour \( s \right... | Yes |
Theorem 24 The expected sojourn times satisfy\n\n\[ \n d\left( s\right) {S}_{x}\left( {s \rightarrow t}\right) = d\left( x\right) {S}_{s}\left( {x \rightarrow t}\right) .\n\] | Proof. Let us define a random walk on the set \( \{ s, t, x\} \) with transition probabilities \( {p}_{st} = {P}_{\mathrm{{esc}}}\left( {s \rightarrow t < x}\right) ,{p}_{sx} = {P}_{\mathrm{{esc}}}\left( {s \rightarrow x < t}\right) ,{p}_{ss} = 1 - {p}_{st} - {p}_{sx} \), and so on. Theorem 22 implies that this new RW ... | No |
Theorem 25 Let \( G \) be a connected graph of order \( n \) . Then\n\n\[ \mathop{\sum }\limits_{{{st} \in E\left( G\right) }}{R}_{\text{eff }}\left( {s, t}\right) = n - 1 \] | First Proof. By Theorem 24, for any two vertices \( t \) and \( x \) we have\n\n\[ \mathop{\sum }\limits_{{s \in \Gamma \left( t\right) }}\frac{{S}_{x}\left( {s \rightarrow t}\right) }{d\left( x\right) } = \mathop{\sum }\limits_{{s \in \Gamma \left( t\right) }}\frac{{S}_{s}\left( {x \rightarrow t}\right) }{d\left( s\ri... | Yes |
Theorem 26 The mixing rate \( \mu \) is precisely \( \lambda = \max \left\{ {{\lambda }_{2},\left| {\lambda }_{n}\right| }\right\} \) . | Proof. Given a distribution \( {\mathbf{p}}_{0} \), set\n\n\[ \n{\mathbf{p}}_{0} = {\alpha \pi } + {\mathbf{p}}_{0}^{\prime } \n\]\n\nwhere\n\n\[ \n\left\langle {{\mathbf{p}}_{0}^{\prime },\pi }\right\rangle = 0. \n\]\n\nThen \( 1 = \left\langle {{\mathbf{p}}_{0},{n\pi }}\right\rangle = \alpha \langle \pi ,{n\pi }\rang... | Yes |
Theorem 27 Let \( G \) be a non-trivial regular graph with conductance \( {\Phi }_{G} \) . Then every LRW on \( G \) is such that\n\n\[ \n{d}_{2}\left( {t + 1}\right) \leq \left( {1 - \frac{1}{4}{\Phi }_{G}^{2}}\right) {d}_{2}\left( t\right) \n\] | Null | No |
Lemma 29 Let \( G = \left( {V, E}\right) \) be a d-regular graph with conductance \( {\Phi }_{G} \), and let \( x : V \rightarrow \mathbb{R}, i \mapsto {x}_{i} \), be such that \( \mathop{\sum }\limits_{{i = 1}}^{n}{x}_{i} = 0 \) . Then\n\n\[\n\mathop{\sum }\limits_{{{ij} \in E}}{\left( {x}_{i} - {x}_{j}\right) }^{2} \... | Proof. Set \( m = \lceil n/2\rceil \) . We shall prove that if \( {y}_{1} \geq {y}_{2} \geq \cdots \geq {y}_{n} \), with \( {y}_{m} = 0 \) , then\n\n\[\n\mathop{\sum }\limits_{{{ij} \in E}}{\left( {y}_{i} - {y}_{j}\right) }^{2} \geq \frac{d}{2}{\Phi }_{G}^{2}\mathop{\sum }\limits_{{i = 1}}^{n}{y}_{i}^{2}\n\]\n\nIt is e... | Yes |
Corollary 30 Every LRW on a regular graph \( G \) of order \( n \) and conductance \( {\Phi }_{G} \) is such that\n\n\[ \n{d}_{1}\left( t\right) \leq {\left( n{d}_{2}\left( t\right) \right) }^{1/2} \leq {\left( 2n\right) }^{1/2}{\left( 1 - \frac{1}{4}{\Phi }_{G}^{2}\right) }^{t/2} \leq {\left( 2n\right) }^{1/2}{\left( ... | Null | No |
Corollary 31 The second eigenvalue of the SRW on a regular graph with conductance \( {\Phi }_{G} \) is at most \( 1 - {\Phi }_{G}^{2} \) . | Proof. With the notation as above, \( \frac{1}{2}\left( {{\lambda }_{2} + 1}\right) \leq 1 - \frac{1}{2}{\Phi }_{G}^{2} \), so \( {\lambda }_{2} \leq 1 - {\Phi }_{G}^{2} \) . | Yes |
Theorem 32 Let \( {\left( {G}_{i}\right) }_{1}^{\infty } \) be a sequence of regular graphs with \( \left| {G}_{i}\right| = {n}_{i} \rightarrow \infty \) . If there is a \( k \in \mathbb{N} \) such that\n\n\[{\Phi }_{{G}_{i}} \geq {\left( \log {n}_{i}\right) }^{-k}\]\n\nfor every sufficiently large \( i \), then the la... | Proof. We have just seen that if \( t \geq 8{\left( \log {n}_{i}\right) }^{{2k} + 1}\log \left( {1/\epsilon }\right) \) then \( {d}_{1}\left( t\right) < \epsilon \) , provided \( {n}_{i} \) is large enough. | Yes |
Theorem 3 Let \( G = \left( {V, E}\right) \) be a multigraph, \( q \geq 1 \) an integer and \( \beta \in \mathbb{R} \) . Then the partition function of the \( q \) -state Potts model on \( G \), with inverse temperature \( \beta \) , is\n\n\[ \n{P}_{G}\left( {q,\beta }\right) = {e}^{-\beta \left| E\right| }{Z}_{G}\left... | Proof. Set\n\n\[ \n{\widetilde{P}}_{G}\left( {q,\beta }\right) = {e}^{\beta \left| E\right| }{P}_{G}\left( {q,\beta }\right) \n\]\n\nso that we have to show that \( {\widetilde{P}}_{G}\left( {q,\beta }\right) = {Z}_{G}\left( {q, v}\right) \) . If \( G = {E}_{n} \) then \( H\left( \omega \right) = 0 \) for every state \... | Yes |
Theorem 4 The partition function of the random cluster model is\n\n\\[ \n{R}_{G}\\left( {q, p}\\right) = {\\left( 1 - p\\right) }^{\\left| E\\right| }{Z}_{G}\\left( {q, v}\\right) ,\n\\]\n\nwhere \\( v = p/\\left( {1 - p}\\right) \\) . | Proof. Set \\( {\\widetilde{R}}_{G}\\left( {q, p}\\right) = {\\left( 1 - p\\right) }^{-\\left| E\\right| }{R}_{G}\\left( {q, p}\\right) \\), so that\n\n\\[ \n{\\widetilde{R}}_{G}\\left( {q, p}\\right) = \\mathop{\\sum }\\limits_{{F \\subset E}}{v}^{\\left| F\\right| }{q}^{k\\langle F\\rangle },\n\\]\n\n(7)\n\nand we ha... | Yes |
Theorem 5 Let \( G \) be a connected graph. Then \( {T}_{G}\left( {1,1}\right) \) is the number of spanning trees of \( G,{T}_{G}\left( {2,1}\right) \) is the number of (edge sets forming) forests in \( G,{T}_{G}\left( {1,2}\right) \) is the number of connected spanning subgraphs, and \( {T}_{G}\left( {2,2}\right) \) i... | Proof. Each of these assertions is immediate from the definition (1) of \( T \) . Thus,\n\n\[ \n{T}_{G}\left( {1,1}\right) = \mathop{\sum }\limits_{{F \subset E\left( G\right) }}{0}^{r\left( G\right) - r\langle F\rangle }{0}^{n\langle F\rangle }\n\]\n\n\[ \n= \left| {\{ F : \;F \subset E\left( G\right), r\langle F\rang... | Yes |
Theorem 6 The chromatic polynomial \( {p}_{G}\left( x\right) \) of a graph \( G \) is\n\n\[ \n{p}_{G}\left( x\right) = {\left( -1\right) }^{r\left( G\right) }{x}^{k\left( G\right) }{T}_{G}\left( {1 - x,0}\right) .\n\] | Proof. The result is immediate from Theorem 2 and the properties of the chromatic polynomial mentioned above. Indeed, \( {p}_{{E}_{n}}\left( x\right) = {x}^{n} \), and for every edge \( e \in E\left( G\right) \) ,\n\n\[ \n{p}_{G}\left( x\right) = \left\{ \begin{array}{ll} \frac{x - 1}{x}{p}_{G - e}\left( x\right) & \te... | Yes |
Theorem 7 Let \( A \) be a finite Abelian group and \( G \) a multigraph. Then\n\n\[ \n{q}_{G}\left( A\right) = {\left( -1\right) }^{n\left( G\right) }{T}_{G}\left( {0,1 - \left| A\right| }\right) .\n\] | Proof. The result is, once again, immediate from Theorem 2 and the properties of the flow polynomial noted above. Indeed, we have shown that \( {q}_{{E}_{n}}\left( A\right) = 1 \) for every \( n \geq 1 \), and if \( e \in E\left( G\right) \) then\n\n\[ \n{q}_{G}\left( A\right) = \left\{ \begin{array}{ll} 0 & \text{ if ... | Yes |
Theorem 8 For every connected graph \( G \) and every vertex \( u \in V\left( G\right) \) we have\n\n\[ \n{a}_{u}\left( G\right) = {T}_{G}\left( {1,0}\right) \n\] | Proof. We shall deduce the assertion from the following four properties of the function \( {a}_{u}\left( G\right) \) .\n\n(i) If \( G = {E}_{1} \) then \( {a}_{u}\left( G\right) = 1 \) .\n\n(ii) If \( G \) contains a loop \( e \) then \( G \) has no acyclic orientation so \( {a}_{u}\left( G\right) = 0 \) .\n\n(iii) Sup... | Yes |
Theorem 9 Let \( G = \left( {V, E}\right) ,0 < p < 1, q = 1 - p \) and \( {E}_{p} \) be as above. Then\n\n\[ \mathbb{P}\left( {r\left\langle {E}_{p}\right\rangle = r\left( G\right) }\right) = {p}^{r\left( G\right) }{q}^{n\left( G\right) }{T}_{G}\left( {1,1/q}\right) . \] | Proof. In view of Theorem 2, it suffices to check that the function \( C\left( G\right) = \) \( \mathbb{P}\left( {r\left\langle {E}_{p}\right\rangle = r\left( G\right) }\right) \) satisfies the conditions of Theorem 2 with \( x = p, y = 1 \) , \( \alpha = 1,\sigma = q \) and \( \tau = p \) .\n\nAlthough this is very ea... | Yes |
Theorem 11 Let \( \mathop{\sum }\limits_{{i, j}}{t}_{ij}{x}^{i}{y}^{j} \) be the Tutte polynomial of a graph \( G \) with at least two edges. Then \( {t}_{10} = {t}_{01} \) . | Null | No |
Theorem 12 (i) For every graph \( G \) the derivative \( {p}_{G}^{\prime }\left( 1\right) \) of the chromatic polynomial \( {p}_{G}\left( x\right) \) satisfies\n\n\[ \n{p}_{G}^{\prime }\left( 1\right) = {\left( -1\right) }^{r\left( G\right) + 1}\theta \left( G\right) .\n\] | Proof. (i) This is immediate from\n\n\[ \n{p}_{G}\left( x\right) = {\left( -1\right) }^{r\left( G\right) }{x}^{k\left( G\right) }{T}_{G}\left( {1 - x,0}\right) = {\left( -1\right) }^{r\left( G\right) }{x}^{k\left( G\right) }\mathop{\sum }\limits_{{i = 0}}^{{n - 1}}{t}_{i0}{\left( 1 - x\right) }^{i}.\n\] | Yes |
Theorem 13 Let \( G \) be a connected graph of order \( n \), with chromatic polynomial \( {p}_{G}\left( x\right) = \mathop{\sum }\limits_{{j = 0}}^{{n - 1}}{\left( -1\right) }^{j}{a}_{j}{x}^{n - j} \) . Then \( {a}_{0} = 1 \leq {a}_{1} \leq \cdots \leq {a}_{l} \) for \( l = \lfloor n/2\rfloor \) . | Proof. We know that\n\n\[ \n{p}_{G}\left( x\right) = {\left( -1\right) }^{n - 1}x\mathop{\sum }\limits_{{i = 0}}^{{n - 1}}{t}_{i0}{\left( -x + 1\right) }^{i}, \n\] \n\nso \n\n\[ \n{\left( -1\right) }^{j}{a}_{j} = {\left( -1\right) }^{n - 1}\mathop{\sum }\limits_{{i = n - j - 1}}^{{n - 1}}{\left( -1\right) }^{n - j - 1}... | Yes |
Theorem 14 Let \( G \) be a 2-connected loopless graph with \( n \) vertices and \( m \) edges, and let \( {T}_{G}\left( {x, y}\right) = \sum {t}_{ij}{x}^{i}{y}^{j} \) . Then \( {t}_{i0} > 0 \) for each \( i,1 \leq i \leq n - 1 \), and \( {t}_{0j} > 0 \) for each \( j,1 \leq j \leq m - n + 1 \) . | Proof. Given a spanning tree \( T \), for \( {E}_{0} \subset E\left( G\right) \) let \( {\gamma }_{T}\left( {E}_{0}\right) \) be the set of chords whose cycles meet \( {E}_{0} \), together with the set of tree-edges whose cuts meet \( {E}_{0} \) :\n\n\[ \n{\gamma }_{T}\left( {E}_{0}\right) = \left\{ {e \in E\left( G\ri... | Yes |
Theorem 15 Let \( G \) be a 2-connected loopless graph that is neither a cycle nor a thick edge. Then \( {t}_{11}\left( G\right) > 0 \) . | Proof. It is easily seen that \( G \) contains a cycle \( C \) and an edge \( {e}_{1} \) joining a vertex of \( C \) to a vertex not on \( C \) . Let \( T \) be a spanning tree that contains \( {e}_{1} \) and all edges of \( C \) except for an edge \( {e}_{2} \), and set \( {E}_{0} = \left\{ {{e}_{1},{e}_{2}}\right\} \... | Yes |
Theorem 16 There is a unique map \( \varphi : \mathcal{L} \rightarrow \mathbb{Z}\left\lbrack {A, B, d}\right\rbrack \) such that\n\n(i) if \( L \) and \( {L}^{\prime } \) are planar homotopic link diagrams then \( \varphi \left( L\right) = \varphi \left( {L}^{\prime }\right) \),\n\n(ii) \( \varphi \left( ○\right) = 1 \... | Proof. It is clear that conditions (i) - (iv) determine a unique map, if there is such a map. Hence all we have to check is that the Kauffman square bracket [.] has properties (i)-(iv). The first three are immediate from the definition.\n\nProperty (iv) is also almost immediate. Indeed, let \( v \) be a crossing of \( ... | Yes |
Lemma 17 The Kauffman bracket is invariant under regular isotopy. | Proof. Let \( B \) and \( d \) be as above, so that \( {AB} = 1 \) and \( d = - {A}^{2} - {A}^{-2} \) and, under these conditions, \( \langle L\rangle \left( A\right) = \left\lbrack L\right\rbrack \left( {A, B, d}\right) \) . First, let us evaluate the effect of a Type II move on the angle bracket by resolving crossing... | Yes |
Theorem 18 The Laurent polynomial \( f\left\lbrack L\right\rbrack = {\left( -A\right) }^{-{3s}\left( L\right) }\langle L\rangle \in \mathbb{Z}\left\lbrack {A,{A}^{-1}}\right\rbrack \) is an invariant of ambient isotopy for unoriented links. | Proof. Since \( s\left( L\right) \) and \( \langle L\rangle \) are invariants of regular isotopy, so is \( f\left\lbrack L\right\rbrack \) . Thus all we have to check is that \( f\left\lbrack L\right\rbrack \) is invariant under Type I Reidemeister moves.\n\nNote first that\n\n\[ \langle \circlearrowleft \rangle = A\la... | Yes |
Theorem 19 The Jones polynomial \( {V}_{L}\left( t\right) \) of an oriented link \( L \) is given by \( {V}_{L}\left( t\right) = f\left\lbrack L\right\rbrack \left( {t}^{-1/4}\right) \), where \( f\left\lbrack L\right\rbrack = {\left( -A\right) }^{-{3w}\left( L\right) }\langle L\rangle \left( A\right) \) . | Proof. Since \( f\left\lbrack \subset \right\rbrack = 1 \), all we have to check is that \( f\left\lbrack L\right\rbrack \left( {t}^{-1/4}\right) \) satisfies (9). By property (iv) of the bracket polynomial, as \( B = {A}^{-1} \) we have\n\n\[ \langle X\rangle = A\langle X\rangle + {A}^{-1}\langle \rangle (\]\n\nand\n\... | Yes |
Theorem 20 The bracket and one-variable Kauffman polynomial of the mirror image \( {L}^{ * } \) of a link diagram \( L \) are\n\n\[ \left\langle {L}^{ * }\right\rangle \left( A\right) = \langle L\rangle \left( {A}^{-1}\right) \]\n\nand\n\n\[ {f}_{{L}^{ * }}\left\lbrack A\right\rbrack = {f}_{L}\left\lbrack {A}^{-1}\righ... | Proof. Note that reversing all the crossings results in swapping \( A \) and \( B \), that is \( A \) and \( {A}^{-1} \), in the expansion of the bracket. Hence \( \left\langle {L}^{ * }\right\rangle \left( A\right) = \langle L\rangle \left( {A}^{-1}\right) \) . Also, \( s\left( {L}^{ * }\right) = - s\left( L\right) \)... | Yes |
Theorem 21 Let \( L \) be a connected alternating oriented link diagram with a A-regions, b B-regions, and writhe w. Then the Jones polynomial of \( L \) is given by the Tutte polynomial of \( {G}^{ + }\left( L\right) \) : | \[ {V}_{L}\left( t\right) = {\left( -1\right) }^{w}{t}^{\left( {b - a + {3w}}\right) /4}{T}_{{G}^{ + }\left( L\right) }\left( {-t, - 1/t}\right) . \] | Yes |
Theorem 22 The number of crossings of a connected alternating link diagram without nugatory crossings is an ambient isotopy invariant. | Proof. Let \( L \) be a connected alternating link diagram with \( m \) crossings, none of which is nugatory. We claim that \( m \) is precisely the breadth of the Laurent polynomial \( {V}_{L}\left( t\right) \), i.e. the difference between the maximum degree and the minimum degree. As the Jones polynomial is ambient i... | Yes |
Theorem 1.1 (Hölder) Suppose \( 1 < p < \infty \) and \( 1 < q < \infty \) are conjugate exponents. If \( f \in {L}^{p} \) and \( g \in {L}^{q} \), then \( {fg} \in {L}^{1} \) and\n\n\[ \parallel {fg}{\parallel }_{{L}^{1}} \leq \parallel f{\parallel }_{{L}^{p}}\parallel g{\parallel }_{{L}^{q}} \] | The proof of the theorem relies on a simple generalized form of the arithmetic-geometric mean inequality: if \( A, B \geq 0 \), and \( 0 \leq \theta \leq 1 \), then\n\n(2)\n\n\[ {A}^{\theta }{B}^{1 - \theta } \leq {\theta A} + \left( {1 - \theta }\right) B. \]\n\nTo establish (2), we observe first that we may assume \(... | Yes |
Theorem 1.2 (Minkowski) If \( 1 \leq p < \infty \) and \( f, g \in {L}^{p} \), then \( f + g \in \) \( {L}^{p} \) and \( \parallel f + g{\parallel }_{{L}^{p}} \leq \parallel f{\parallel }_{{L}^{p}} + \parallel g{\parallel }_{{L}^{p}} \) . | Proof. The case \( p = 1 \) is obtained by integrating \( \left| {f\left( x\right) + g\left( x\right) }\right| \leq \) \( \left| {f\left( x\right) }\right| + \left| {g\left( x\right) }\right| \) . When \( p > 1 \), we may begin by verifying that \( f + g \in {L}^{p} \) , when both \( f \) and \( g \) belong to \( {L}^{... | Yes |
Proposition 1.4 If \( X \) has finite positive measure, and \( {p}_{0} \leq {p}_{1} \), then \( {L}^{{p}_{1}}\left( X\right) \subset {L}^{{p}_{0}}\left( X\right) \) and\n\n\[ \frac{1}{\mu {\left( X\right) }^{1/{p}_{0}}}\parallel f{\parallel }_{{L}^{{p}_{0}}} \leq \frac{1}{\mu {\left( X\right) }^{1/{p}_{1}}}\parallel f{... | We may assume that \( {p}_{1} > {p}_{0} \) . Suppose \( f \in {L}^{{p}_{1}} \), and set \( F = {\left| f\right| }^{{p}_{0}} \) , \( G = 1, p = {p}_{1}/{p}_{0} > 1 \), and \( 1/p + 1/q = 1 \), in Hölder’s inequality applied to \( F \) and \( G \) . This yields\n\n\[ \parallel f{\parallel }_{{L}^{{p}_{0}}}^{{p}_{0}} \leq... | Yes |
Proposition 1.5 If \( X = \mathbb{Z} \) is equipped with counting measure, then the reverse inclusion holds, namely \( {L}^{{p}_{0}}\left( \mathbb{Z}\right) \subset {L}^{{p}_{1}}\left( \mathbb{Z}\right) \) if \( {p}_{0} \leq {p}_{1} \) . Moreover, \( \parallel f{\parallel }_{{L}^{{p}_{1}}} \leq \parallel f{\parallel }_... | Indeed, if \( f = \{ f\left( n\right) {\} }_{n \in \mathbb{Z}} \), then \( \sum {\left| f\left( n\right) \right| }^{{p}_{0}} = \parallel f{\parallel }_{{L}^{{p}_{0}}}^{{p}_{0}} \), and \( \mathop{\sup }\limits_{n}\left| {f\left( n\right) }\right| \leq \) \( \parallel f{\parallel }_{{L}^{{p}_{0}}} \) . However\n\n\[ \su... | Yes |
Theorem 2.1 The vector space \( {L}^{\infty } \) equipped with \( \parallel \cdot {\parallel }_{{L}^{\infty }} \) is a complete vector space. | Null | No |
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