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Corollary 6.13.4 Let \( X \) be a vertex-transitive graph on \( n \) vertices with chromatic number three. If \( n \) is not a multiple of three, then \( X \) is triangle-free. | Proof. Since \( X \) is 3-colourable, it has a homomorphism onto \( {K}_{3} \) . If \( X \) contained a triangle, then the core of \( X \) would be a triangle and \( n \) would be a multiple of three, contradicting the hypothesis. Therefore, \( X \) has no triangles. | Yes |
Theorem 6.13.5 If \( X \) is a connected 2-arc transitive nonbipartite graph, then \( X \) is a core. | Proof. Since \( X \) is not bipartite, it contains an odd cycle; since \( X \) is 2-arc transitive, each 2-arc lies in a shortest odd cycle. | No |
Theorem 6.14.1 If \( X \) is a connected arc-transitive nonbipartite cubic graph, then \( X \) is a core. | Proof. Let \( C \) be a shortest odd cycle in \( X \), and let \( x \) be a vertex in \( C \) with three neighbours \( {x}_{1},{x}_{2} \), and \( {x}_{3} \), where \( {x}_{1} \) and \( {x}_{2} \) are in \( C \) . If \( G \) is the automorphism group of \( X \), then the vertex stabilizer \( {G}_{x} \) contains an eleme... | Yes |
Theorem 6.14.3 If \( X \) is a connected vertex-transitive cubic graph, then \( {X}^{ \bullet } \) is \( {K}_{2} \), an odd cycle, or \( X \) itself. | Proof. The proof of this is left as Exercise 42. | No |
Lemma 7.2.2 If \( X \) is vertex transitive, then\n\n\[{\omega }^{ * }\left( X\right) = \frac{\left| V\left( X\right) \right| }{\alpha \left( X\right) }\]\n\nand \( \alpha {\left( X\right) }^{-1}\mathbf{1} \) is a fractional clique with this weight. | Proof. Suppose \( g \) is a nonzero fractional clique of \( X \) . Then \( g \) is a function on \( V\left( X\right) \) . If \( \gamma \in \operatorname{Aut}\left( X\right) \), define the function \( {g}^{\gamma } \) by\n\n\[{g}^{\gamma }\left( x\right) = g\left( {x}^{\gamma }\right)\]\n\nThen \( {g}^{\gamma } \) is ag... | Yes |
Lemma 7.2.2 If \( X \) is vertex transitive, then\n\n\[{\omega }^{ * }\left( X\right) = \frac{\left| V\left( X\right) \right| }{\alpha \left( X\right) }\]\n\nand \( \alpha {\left( X\right) }^{-1}\mathbf{1} \) is a fractional clique with this weight. | Proof. Suppose \( g \) is a nonzero fractional clique of \( X \) . Then \( g \) is a function on \( V\left( X\right) \) . If \( \gamma \in \operatorname{Aut}\left( X\right) \), define the function \( {g}^{\gamma } \) by\n\n\[{g}^{\gamma }\left( x\right) = g\left( {x}^{\gamma }\right)\]\n\nThen \( {g}^{\gamma } \) is ag... | Yes |
Lemma 7.3.1 If a graph \( X \) has a fractional colouring \( f \) of weight \( w \), then it has a fractional colouring \( {f}^{\prime } \) with weight no greater than \( w \) such that \( B{f}^{\prime } = 1 \) . | Proof. If \( {Bf} \neq \mathbf{1} \), then we will show that we can perturb \( f \) into a function \( {f}^{\prime } \) of weight no greater than \( f \) such that \( B{f}^{\prime } \) has fewer entries not equal to one. The result then follows immediately by induction.\n\nSuppose that some entry of \( {Bf} \) is great... | Yes |
Theorem 7.4.1 If there is a homomorphism from \( X \) to \( Y \) and \( f \) is a fractional colouring of \( Y \), then the lift \( \widehat{f} \) of \( f \) is a fractional colouring of \( X \) with weight equal to the weight of \( f \) . The support of \( \widehat{f} \) consists of the preimages of the independent se... | Proof. If \( u \in V\left( X\right) \), then\n\n\[ \mathop{\sum }\limits_{{T \in \mathcal{I}\left( {X, u}\right) }}\widehat{f}\left( T\right) = \mathop{\sum }\limits_{{S : u \in {\varphi }^{-1}\left( S\right) }}f\left( S\right) \]\n\n\[ = \mathop{\sum }\limits_{{S \in \mathcal{I}\left( {Y,\varphi \left( u\right) }\righ... | Yes |
Corollary 7.4.2 If there is a homomorphism from \( X \) to \( Y \), then \( {\chi }^{ * }\left( X\right) \leq \) \( {\chi }^{ * }\left( Y\right) \) . | If there is an independent set in the support of \( f \) that does not intersect \( \varphi \left( X\right) \), then its preimage is the empty set. In this situation \( \widehat{f}\left( \varnothing \right) \neq 0 \) , and there is a fractional colouring that agrees with \( \widehat{f} \) on all nonempty independent se... | No |
Lemma 7.4.4 If \( X \) is vertex transitive, then \( {\chi }^{ * }\left( X\right) \leq \left| {V\left( X\right) }\right| /\alpha \left( X\right) \) . | Proof. We saw in Section 7.1 that \( {\chi }^{ * }\left( {K}_{v : r}\right) \leq v/r \) . If \( X \) is vertex transitive, then by Theorem 3.9.1 and the remarks following its proof, it is a retract of a Cayley graph \( Y \) where \( \left| {V\left( Y\right) }\right| /\alpha \left( Y\right) = \left| {V\left( X\right) }\... | Yes |
Lemma 7.4.4 If \( X \) is vertex transitive, then \( {\chi }^{ * }\left( X\right) \leq \left| {V\left( X\right) }\right| /\alpha \left( X\right) \) . | Proof. We saw in Section 7.1 that \( {\chi }^{ * }\left( {K}_{v : r}\right) \leq v/r \) . If \( X \) is vertex transitive, then by Theorem 3.9.1 and the remarks following its proof, it is a retract of a Cayley graph \( Y \) where \( \left| {V\left( Y\right) }\right| /\alpha \left( Y\right) = \left| {V\left( X\right) }\... | Yes |
Theorem 7.4.5 For any graph \( X \) we have\n\n\[ \n{\chi }^{ * }\left( X\right) = \min \left\{ {v/r : X \rightarrow {K}_{v : r}}\right\} \n\] | Proof. We have already seen that \( {\chi }^{ * }\left( {K}_{v : r}\right) \leq v/r \), and so by Corollary 7.4.2, it follows that if \( X \) has a homomorphism into \( {K}_{v : r} \), then it has a fractional colouring with weight at most \( v/r \) .\n\nConversely, suppose that \( X \) is a graph with fractional chrom... | Yes |
Lemma 7.5.1 For any graph \( X \) we have \( {\omega }^{ * }\left( X\right) \leq {\chi }^{ * }\left( X\right) \) . | Proof. Suppose that \( f \) is a fractional colouring and \( g \) a fractional clique of \( X \) . Then\n\n\[ \n{\mathbf{1}}^{T}f - {g}^{T}\mathbf{1} = {\mathbf{1}}^{T}f - {g}^{T}{Bf} + {g}^{T}{Bf} - {g}^{T}\mathbf{1} \n\]\n\n\[ \n= \left( {{\mathbf{1}}^{T} - {g}^{T}B}\right) f + {g}^{T}\left( {{Bf} - \mathbf{1}}\right... | Yes |
Corollary 7.5.2 If \( X \) is a vertex-transitive graph, then\n\n\[{\omega }^{ * }\left( X\right) = {\chi }^{ * }\left( X\right) = \frac{\left| V\left( X\right) \right| }{\alpha \left( X\right) }.\] | Proof. Lemma 7.2.1, Lemma 7.5.1, and Lemma 7.4.4 yield that\n\n\[ \frac{\left| V\left( X\right) \right| }{\alpha \left( X\right) } \leq {\omega }^{ * }\left( X\right) \leq {\chi }^{ * }\left( X\right) \leq \frac{\left| V\left( X\right) \right| }{\alpha \left( X\right) } \]\n\nand so the result follows. | Yes |
Corollary 7.5.3 For any graph \( X \) we have\n\n\[{\chi }^{ * }\left( X\right) \geq \frac{\left| V\left( X\right) \right| }{\alpha \left( X\right) }\] | Proof. Use Lemma 7.2.1. | No |
Lemma 7.5.4 Let \( X \) and \( Y \) be vertex-transitive graphs with the same fractional chromatic number, and suppose \( \varphi \) is a homomorphism from \( X \) to \( Y \) . If \( S \) is a maximum independent set in \( Y \), then \( {\varphi }^{-1}\left( S\right) \) is a maximum independent set in \( X \) . | Proof. Since \( X \) and \( Y \) are vertex transitive, \[ \frac{\left| V\left( X\right) \right| }{\alpha \left( X\right) } = {\chi }^{ * }\left( X\right) = {\chi }^{ * }\left( Y\right) = \frac{\left| V\left( Y\right) \right| }{\alpha \left( Y\right) }. \] Let \( f \) be a fractional colouring of weight \( {\chi }^{ * ... | Yes |
Lemma 7.6.1 Let \( X \) be a minimally imperfect graph. Then any independent set is disjoint from at least one big clique. | Proof. Let \( S \) be an independent set in the minimally imperfect graph \( X \) . Then \( X \smallsetminus S \) is perfect, and therefore \( \chi \left( {X \smallsetminus S}\right) = \omega \left( {X \smallsetminus S}\right) \) . If \( S \) meets each big clique in at least one vertex, it follows that \( \omega \left... | Yes |
Lemma 7.6.2 Each vertex of \( X \) lies in exactly \( \alpha \) members of \( \mathcal{S} \), and any big clique of \( X \) is disjoint from exactly one member of \( \mathcal{S} \) . | Proof. We leave the first claim as an exercise.\n\nLet \( K \) be a big clique of \( X \), let \( v \) be an arbitrary vertex of \( X \), and suppose that \( X \smallsetminus v \) is coloured with \( \omega \) colours. Then \( K \) has at most one vertex in each colour class, and so either \( v \notin K \) and \( K \) ... | No |
Theorem 7.6.4 The complement of a perfect graph is perfect. | Proof. For any graph \( X \) we have the trivial bound \( \left| {V\left( X\right) }\right| \leq \chi \left( X\right) \alpha \left( X\right) \) , and so for perfect graphs we have \( \left| {V\left( X\right) }\right| \leq \alpha \left( X\right) \omega \left( X\right) \) . Since \( J - I \) is invertible, the previous l... | Yes |
Theorem 7.6.4 The complement of a perfect graph is perfect. | Proof. For any graph \( X \) we have the trivial bound \( \left| {V\left( X\right) }\right| \leq \chi \left( X\right) \alpha \left( X\right) \) , and so for perfect graphs we have \( \left| {V\left( X\right) }\right| \leq \alpha \left( X\right) \omega \left( X\right) \) .\n\nSince \( J - I \) is invertible, the previou... | Yes |
Lemma 7.7.1 For \( v \geq {2r} \), an independent set in \( C\left( {v, r}\right) \) has size at most \( r \) . Moreover, an independent set of size \( r \) consists of the vertices that contain a given element of \( \{ 1,\ldots, v\} \) . | Proof. Suppose that \( S \) is an independent set in \( C\left( {v, r}\right) \) . Since \( C\left( {v, r}\right) \) is vertex transitive, we may assume that \( S \) contains the \( r \) -set \( \beta = \{ 1,\ldots, r\} \) , Let \( {S}_{1} \) and \( {S}_{r} \) be the \( r \) -sets in \( S \) that contain the points 1 a... | Yes |
Corollary 7.7.3 For \( v \geq {2r} \), the fractional chromatic number of the Kneser graph \( {K}_{v : r} \) is \( v/r \) . | Proof. Since \( C\left( {v, r}\right) \) is a subgraph of \( {K}_{v : r} \), it follows that\n\n\[ \frac{v}{r} = {\chi }^{ * }\left( {C\left( {v, r}\right) }\right) \leq {\chi }^{ * }\left( {K}_{v : r}\right) \]\n\nand we have already seen that \( {\chi }^{ * }\left( {K}_{v : r}\right) \leq v/r \) . | Yes |
Corollary 7.7.4 If \( v > {2r} \), then the shortest odd cycle in \( {K}_{v : r} \) has length at least \( v/\left( {v - {2r}}\right) \) . | Proof. If the odd cycle \( {C}_{{2m} + 1} \) is a subgraph of \( {K}_{v : r} \), then\n\n\[ 2 + \frac{1}{m} = {\chi }^{ * }\left( {C}_{{2m} + 1}\right) \leq \frac{v}{r} \]\n\nwhich implies that \( m \geq r/\left( {v - {2r}}\right) \), and hence that \( {2m} + 1 \geq v/\left( {v - {2r}}\right) .▱ \) | Yes |
Corollary 7.8.2 The automorphism group of \( {K}_{v : r} \) is isomorphic to the symmetric group \( \operatorname{Sym}\left( v\right) \) . | Proof. Let \( X \) denote \( {K}_{v : r} \) and let \( X\left( i\right) \) denote the maximum independent set consisting of all the \( r \) -sets containing the point \( i \) from the underlying set \( \Omega \) . Any automorphism of \( X \) must permute the maximum independent sets of \( X \), and by the Erdős-Ko-Rado... | Yes |
Theorem 7.9.1 If \( v > {2r} \), then \( {K}_{v : r} \) is a core. | Proof. Let \( X \) denote \( {K}_{v : r} \), and let \( X\left( i\right) \) denote the maximum independent set consisting of all the \( r \) -sets containing the point \( i \) from the underlying set \( \Omega \) . Let \( \varphi \) be a homomorphism from \( X \) to \( X \) . We will show that it is onto. If \( \beta =... | Yes |
Theorem 7.9.2 If \( v \geq {2r} \) and \( r \geq 2 \), there is a homomorphism from \( {K}_{v : r} \) to \( {K}_{v - 2 : r - 1} \) . | Proof. If \( v = {2r} \), then \( {K}_{v : r} = \left( \begin{matrix} {2r} - 1 \\ r - 1 \end{matrix}\right) {K}_{2} \), which admits a homomorphism into any graph with an edge. So we assume \( v > {2r} \), and that the underlying set \( \Omega \) is equal to \( \{ 1,\ldots, v\} \) . We can easily find a homomorphism \(... | Yes |
Lemma 7.9.3 Suppose that \( v > {2r} \) and \( v/r = w/s \) . There is a homomorphism from \( {K}_{v : r} \) to \( {K}_{w : s} \) if and only if \( r \) divides \( s \) . | Proof. Suppose \( r \) divides \( s \) ; we may assume \( s = {mr} \) and \( w = {mv} \) . Let \( W \) be a fixed set of size \( w \) and let \( \pi \) be a partition of it into \( v \) cells of size \( m \) . Then the \( s \) -subsets of \( W \) that are the union of \( r \) cells of \( \pi \) induce a subgraph of \( ... | Yes |
Theorem 7.10.1 (Hilton-Milner) If \( v \geq {2r} \), the maximum size of an independent set in \( {K}_{v : r} \) with no centre is\n\n\[{\mathrm{h}}_{v, r} = 1 + \left( \begin{array}{l} v - 1 \\ r - 1 \end{array}\right) - \left( \begin{matrix} v - r - 1 \\ r - 1 \end{matrix}\right) . | Proof. Suppose that \( f \) is a homomorphism from \( X = {K}_{v : r} \) to \( Y = {K}_{w : \ell } \) . Consider the preimages \( {f}^{-1}\left( {Y\left( i\right) }\right) \) of all the maximum independent sets of \( Y \), and suppose that two of them, say \( {f}^{-1}\left( {Y\left( i\right) }\right) \) and \( {f}^{-1}... | Yes |
Lemma 7.10.2 Suppose there is a homomorphism from \( {K}_{v : r} \) to \( {K}_{w : \ell } \) . If\n\n\[ \ell \left( \begin{array}{l} v \\ r \end{array}\right) > v\left( \begin{array}{l} v - 1 \\ r - 1 \end{array}\right) + \left( {w - v}\right) {\mathrm{h}}_{v, r} \]\n\nthen there is a homomorphism from \( {K}_{v - 1 : ... | Proof. Suppose that \( f \) is a homomorphism from \( X = {K}_{v : r} \) to \( Y = {K}_{w : \ell } \) . Consider the preimages \( {f}^{-1}\left( {Y\left( i\right) }\right) \) of all the maximum independent sets of \( Y \), and suppose that two of them, say \( {f}^{-1}\left( {Y\left( i\right) }\right) \) and \( {f}^{-1}... | Yes |
Lemma 7.11.2 Let \( \mathcal{C} \) be a collection of closed convex subsets of the unit sphere in \( {\mathbb{R}}^{n} \) . Let \( X \) be the graph with the elements of \( \mathcal{C} \) as its vertices, with two elements adjacent if they are disjoint. If for each unit vector a the open half-space \( H\left( a\right) \... | Proof. Suppose \( X \) has been coloured with the \( n \) colours \( \{ 1,\ldots, n\} \) . For \( i \in \{ 1,\ldots, n\} \), let \( {C}_{i} \) be the set of vectors \( a \) on the unit sphere such that \( H\left( a\right) \) contains a vertex of colour \( i \) . If \( S \in V\left( X\right) \), then the set of vectors ... | Yes |
Theorem 7.11.4 \( \chi \left( {K}_{v : r}\right) = v - {2r} + 2 \) . | Proof. We have already seen that \( v - {2r} + 2 \) is an upper bound on \( \chi \left( {K}_{v : r}\right) \) ; we must show that it is also a lower bound.\n\nAssume that \( \Omega = \left\{ {{x}_{1},\ldots ,{x}_{v}}\right\} \) is a set of \( v \) points in \( {\mathbb{R}}^{v - {2r} + 1} \) such that each open half-spa... | Yes |
Theorem 7.11.4 \( \chi \left( {K}_{v : r}\right) = v - {2r} + 2 \) . | Proof. We have already seen that \( v - {2r} + 2 \) is an upper bound on \( \chi \left( {K}_{v : r}\right) \) ; we must show that it is also a lower bound.\n\nAssume that \( \Omega = \left\{ {{x}_{1},\ldots ,{x}_{v}}\right\} \) is a set of \( v \) points in \( {\mathbb{R}}^{v - {2r} + 1} \) such that each open half-spa... | Yes |
Theorem 7.13.1 (Welzl) Let \( X \) be a graph such that \( \chi \left( X\right) \geq 3 \), and let \( Z \) be a graph such that \( X \rightarrow Z \) but \( Z \nrightarrow X \) . Then there is a graph \( Y \) such that \( X \rightarrow Y \) and \( Y \rightarrow Z \), but \( Z \nrightarrow Y \) and \( Y \rightarrow X \)... | Proof. Since \( X \) is not empty or bipartite, any homomorphism from \( X \) to \( Z \) must map \( X \) into a nonbipartite component of \( Z \) . If we have homomorphisms \( X \rightarrow Y \) and \( Y \rightarrow Z \), it follows that the image of \( Y \) must be contained in a nonbipartite component of \( Z \) . S... | Yes |
Lemma 7.14.1 For any graph \( X \), we have \( {\alpha }_{r}\left( X\right) = \alpha \left( {X▱{K}_{r}}\right) \) . | Proof. Suppose that \( S \) is an independent set in \( X▱{K}_{r} \). If \( v \in V\left( {K}_{r}\right) \), then the set \( {S}_{v} \), defined by\n\n\[ \n{S}_{v} = \{ u \in V\left( X\right) : \left( {u, v}\right) \in S\} \n\]\n\nis an independent set in \( X \). Any two distinct vertices of \( X▱{K}_{r} \) with the s... | Yes |
Lemma 7.14.2 If \( Y \) is vertex transitive and there is a homomorphism from \( X \) to \( Y \), then\n\n\[ \frac{\left| V\left( X\right) \right| }{{\alpha }_{r}\left( X\right) } \leq \frac{\left| V\left( Y\right) \right| }{{\alpha }_{r}\left( Y\right) } \] | Proof. If there is a homomorphism from \( X \) to \( Y \), then there is a homomorphism from \( X▱{K}_{r} \) to \( Y▱{K}_{r} \) . Therefore, \( {\chi }^{ * }\left( {X▱{K}_{r}}\right) \leq {\chi }^{ * }\left( {Y▱{K}_{r}}\right) \). \n\nUsing Corollary 7.5.3 and Corollary 7.5.2 in turn, we see that\n\n\[ \frac{\left| V\l... | Yes |
Lemma 7.15.1 For graphs \( X \) and \( Y \) , \n\n\[ \n\left| {\operatorname{Hom}\left( {X * Y,{K}_{n}}\right) }\right| = \left| {\operatorname{Hom}\left( {Y,{\mathcal{C}}_{n}\left( X\right) }\right) }\right| \n\] | Proof. Exercise. | No |
Theorem 7.15.2 \( {\mathcal{C}}_{n}\left( {K}_{r}\right) = {K}_{n : r}\left\lbrack \overline{{K}_{r!}}\right\rbrack \) . | Proof. Each vertex of \( {\mathcal{C}}_{n}\left( {K}_{r}\right) \) is an \( n \) -colouring of \( {K}_{r} \), and so its image (as a function) is a set of \( r \) distinct colours. Partitioning the vertices of \( {\mathcal{C}}_{n}\left( {K}_{r}\right) \) according to their images gives \( \left( \begin{array}{l} n \\ r... | Yes |
Lemma 7.15.5 The number of v-colourings of the graph \( {C}_{n}\left\lbrack {K}_{r}\right\rbrack \) is equal to\n\n\[ \left| {\operatorname{Hom}\left( {{C}_{n},{K}_{v : r}\left\lbrack \overline{{K}_{r!}}\right\rbrack }\right) }\right| \] | Proof. The lexicographic product \( {C}_{n}\left\lbrack {K}_{r}\right\rbrack \) is equal to the strong product \( {K}_{r} * {C}_{n} \), and therefore we have\n\n\[ \left| {\operatorname{Hom}\left( {{C}_{n}\left\lbrack {K}_{r}\right\rbrack ,{K}_{v}}\right) }\right| = \left| {\operatorname{Hom}\left( {{K}_{r} * {C}_{n},{... | Yes |
Lemma 8.1.1 Let \( X \) and \( Y \) be directed graphs on the same vertex set. Then they are isomorphic if and only if there is a permutation matrix \( P \) such that \( {P}^{T}A\left( X\right) P = A\left( Y\right) \) . | Since permutation matrices are orthogonal, \( {P}^{T} = {P}^{-1} \), and so if \( X \) and \( Y \) are isomorphic directed graphs, then \( A\left( X\right) \) and \( A\left( Y\right) \) are similar matrices. The characteristic polynomial of a matrix \( A \) is the polynomial\n\n\[ \phi \left( {A, x}\right) = \det \left... | Yes |
Lemma 8.1.2 Let \( X \) be a directed graph with adjacency matrix \( A \) . The number of walks from \( u \) to \( v \) in \( X \) with length \( r \) is \( {\left( {A}^{r}\right) }_{uv} \) . | Proof. This is easily proved by induction on \( r \), as you are invited to do. \( ▱ \) | No |
Corollary 8.1.3 Let \( X \) be a graph with e edges and \( t \) triangles. If \( A \) is the adjacency matrix of \( X \), then\n\n(a) \( \operatorname{tr}A = 0 \) ,\n\n(b) \( \operatorname{tr}{A}^{2} = {2e} \) ,\n\n(c) \( \operatorname{tr}{A}^{3} = {6t} \) . | Since the trace of a square matrix is also equal to the sum of its eigenvalues, and the eigenvalues of \( {A}^{r} \) are the \( r \) th powers of the eigenvalues of \( A \), we see that \( \operatorname{tr}{A}^{r} \) is determined by the spectrum of \( A \) . Therefore, the spectrum of a graph \( X \) determines at lea... | No |
Lemma 8.2.4 If \( C \) and \( D \) are matrices such that \( {CD} \) and \( {DC} \) are both defined, then \( \det \left( {I - {CD}}\right) = \det \left( {I - {DC}}\right) \) . | Proof. If\n\n\[ \nX = \left( \begin{matrix} I & C \\ D & I \end{matrix}\right) ,\;Y = \left( \begin{matrix} I & 0 \\ - D & I \end{matrix}\right) , \n\] \n\nthen \n\n\[ \n{XY} = \left( \begin{matrix} I - {CD} & C \\ 0 & I \end{matrix}\right) ,\;{YX} = \left( \begin{matrix} I & C \\ 0 & I - {DC} \end{matrix}\right) , \n\... | Yes |
Lemma 8.2.4 If \( C \) and \( D \) are matrices such that \( {CD} \) and \( {DC} \) are both defined, then \( \det \left( {I - {CD}}\right) = \det \left( {I - {DC}}\right) \) . | Proof. If\n\n\[ \nX = \left( \begin{matrix} I & C \\ D & I \end{matrix}\right) ,\;Y = \left( \begin{matrix} I & 0 \\ - D & I \end{matrix}\right) , \n\] \n\nthen \n\n\[ \n{XY} = \left( \begin{matrix} I - {CD} & C \\ 0 & I \end{matrix}\right) ,\;{YX} = \left( \begin{matrix} I & C \\ 0 & I - {DC} \end{matrix}\right) , \n\... | Yes |
Lemma 8.2.5 Let \( X \) be a regular graph of valency \( k \) with \( n \) vertices and e edges and let \( L \) be the line graph of \( X \) . Then\n\n\[ \phi \left( {L, x}\right) = {\left( x + 2\right) }^{e - n}\phi \left( {X, x - k + 2}\right) . \] | Proof. Substituting \( C = {x}^{-1}{B}^{T} \) and \( D = B \) into the previous lemma we get\n\n\[ \det \left( {{I}_{e} - {x}^{-1}{B}^{T}B}\right) = \det \left( {{I}_{n} - {x}^{-1}B{B}^{T}}\right) \]\n\nwhence\n\n\[ \det \left( {x{I}_{e} - {B}^{T}B}\right) = {x}^{e - n}\det \left( {x{I}_{n} - B{B}^{T}}\right) . \]\n\nN... | Yes |
Lemma 8.3.3 Let \( X \) and \( Y \) be dual plane graphs, and let \( \sigma \) be an orientation of \( X \) . If \( D \) and \( E \) are the incidence matrices of \( {X}^{\sigma } \) and \( {Y}^{\sigma } \), then \( D{E}^{T} = 0 \) . | Proof. If \( u \) is an edge of \( X \) and \( F \) is a face, there are exactly two edges on \( u \) and in \( F \) . Denote them by \( g \) and \( h \) and assume, for convenience, that \( g \) precedes \( h \) as we go clockwise around \( F \) . Then the \( {uF} \) -entry of \( D{E}^{T} \) is equal to\n\n\[ \n{D}_{u... | Yes |
Lemma 8.4.1 Let \( A \) be a real symmetric matrix. If \( u \) and \( v \) are eigenvectors of \( A \) with different eigenvalues, then \( u \) and \( v \) are orthogonal. | Proof. Suppose that \( {Au} = {\lambda u} \) and \( {Av} = {\tau v} \) . As \( A \) is symmetric, \( {u}^{T}{Av} = \) \( {\left( {v}^{T}Au\right) }^{T} \) . However, the left-hand side of this equation is \( \tau {u}^{T}v \) and the right-hand side is \( \lambda {u}^{T}v \), and so if \( \tau \neq \lambda \), it must b... | Yes |
Lemma 8.4.2 The eigenvalues of a real symmetric matrix \( A \) are real numbers. | Proof. Let \( u \) be an eigenvector of \( A \) with eigenvalue \( \lambda \) . Then by taking the complex conjugate of the equation \( {Au} = {\lambda u} \) we get \( A\bar{u} = \bar{\lambda }\bar{u} \), and so \( \bar{u} \) is also an eigenvector of \( A \) . Now, by definition an eigenvector is not zero, so \( {u}^{... | No |
Lemma 8.4.3 Let \( A \) be a real symmetric \( n \times n \) matrix. If \( U \) is an \( A \) - invariant subspace of \( {\mathbb{R}}^{n} \), then \( {U}^{ \bot } \) is also \( A \) -invariant. | Proof. For any two vectors \( u \) and \( v \), we have\n\n\[ \n{v}^{T}\left( {Au}\right) = {\left( Av\right) }^{T}u \n\]\n\nIf \( u \in U \), then \( {Au} \in U \) ; hence if \( v \in {U}^{ \bot } \), then \( {v}^{T}{Au} = 0 \) . Consequently, \( {\left( Av\right) }^{T}u = 0 \) whenever \( u \in U \) and \( v \in {U}^... | Yes |
Lemma 8.4.4 Let \( A \) be an \( n \times n \) real symmetric matrix. If \( U \) is a nonzero \( A \) -invariant subspace of \( {\mathbb{R}}^{n} \), then \( U \) contains a real eigenvector of \( A \) . | Proof. Let \( R \) be a matrix whose columns form an orthonormal basis for \( U \) . Then, because \( U \) is \( A \) -invariant, \( {AR} = {RB} \) for some square matrix \( B \) . Since \( {R}^{T}R = I \), we have\n\n\[ \n{R}^{T}{AR} = {R}^{T}{RB} = B, \n\]\n\nwhich implies that \( B \) is symmetric, as well as real. ... | Yes |
Theorem 8.4.5 Let \( A \) be a real symmetric \( n \times n \) matrix. Then \( {\mathbb{R}}^{n} \) has an orthonormal basis consisting of eigenvectors of \( A \) . | Proof. Let \( \left\{ {{u}_{1},\ldots ,{u}_{m}}\right\} \) be an orthonormal (and hence linearly independent) set of \( m < n \) eigenvectors of \( A \), and let \( M \) be the subspace that they span. Since \( A \) has at least one eigenvector, \( m \geq 1 \) . The subspace \( M \) is \( A \) -invariant, and hence \( ... | Yes |
Corollary 8.4.6 If \( A \) is an \( n \times n \) real symmetric matrix, then there are matrices \( L \) and \( D \) such that \( {L}^{T}L = L{L}^{T} = I \) and \( {LA}{L}^{T} = D \), where \( D \) is the diagonal matrix of eigenvalues of \( A \) . | Proof. Let \( L \) be the matrix whose rows are an orthonormal basis of eigenvectors of \( A \) . We leave it as an exercise to show that \( L \) has the stated properties. | No |
Lemma 8.5.1 Let \( X \) be a \( k \) -regular graph on \( n \) vertices with eigenvalues \( k,{\theta }_{2},\ldots ,{\theta }_{n} \) . Then \( X \) and its complement \( \bar{X} \) have the same eigenvectors, and the eigenvalues of \( \bar{X} \) are \( n - k - 1, - 1 - {\theta }_{2},\ldots , - 1 - {\theta }_{n} \) . | Proof. The adjacency matrix of the complement \( \bar{X} \) is given by\n\n\[ A\left( \bar{X}\right) = J - I - A\left( X\right) \]\n\nwhere \( J \) is the all-ones matrix. Let \( \left\{ {\mathbf{1},{u}_{2},\ldots ,{u}_{n}}\right\} \) be an orthonormal basis of eigenvectors of \( A\left( X\right) \) . Then 1 is an eige... | Yes |
Lemma 8.6.1 If \( A \) is a positive semidefinite matrix, then there is a matrix \( B \) such that \( A = {B}^{T}B \) . | Proof. Since \( A \) is symmetric, there is a matrix \( L \) such that\n\n\[ A = {L}^{T}{\Lambda L} \]\n\nwhere \( \Lambda \) is the diagonal matrix with \( i \) th entry equal to the \( i \) th eigenvalue of \( A \) . Since \( A \) is positive semidefinite, the entries of \( \Lambda \) are nonnegative, and so there is... | Yes |
Lemma 8.6.2 If \( L \) is a line graph, then \( {\theta }_{\min }\left( L\right) \geq - 2 \) . | Proof. If \( L \) is the line graph of \( X \) and \( B \) is the incidence matrix of \( X \), we have\n\n\[ A\left( L\right) + {2I} = {B}^{T}B. \]\n\nSince \( {B}^{T}B \) is positive semidefinite, its eigenvalues are nonnegative and all eigenvalues of \( {B}^{T}B - {2I} \) are at least -2. | Yes |
Lemma 8.6.3 Let \( Y \) be an induced subgraph of \( X \) . Then\n\n\[{\theta }_{\min }\left( X\right) \leq {\theta }_{\min }\left( Y\right) \leq {\theta }_{\max }\left( Y\right) \leq {\theta }_{\max }\left( X\right)\] | Proof. Let \( A \) be the adjacency matrix of \( X \) and abbreviate \( {\theta }_{\max }\left( X\right) \) to \( \theta \) . The matrix \( {\theta I} - A \) has only nonnegative eigenvalues, and is therefore positive semidefinite. Let \( f \) be any vector that is zero on the vertices of \( X \) not in \( Y \), and le... | Yes |
Lemma 8.7.2 Let \( A \) be an \( n \times n \) nonnegative irreducible matrix and let \( \rho \) be the greatest real number such that \( A \) has a \( \rho \) -subharmonic vector. If \( B \) is an \( n \times n \) matrix such that \( \left| B\right| \leq A \) and \( {Bx} = {\theta x} \), then \( \left| \theta \right| ... | Proof. If \( {Bx} = {\theta x} \), then\n\n\[ \left| \theta \right| \left| x\right| = \left| {\theta x}\right| = \left| {Bx}\right| \leq \left| B\right| \left| x\right| \leq A\left| x\right| .\n\]\n\nHence \( \left| x\right| \) is \( \left| \theta \right| \) -subharmonic for \( A \), and so \( \left| \theta \right| \le... | Yes |
Lemma 8.7.3 Let \( A \) be a nonnegative irreducible \( n \times n \) matrix with spectral radius \( \rho \) . Then \( \rho \) is a simple eigenvalue of \( A \), and if \( x \) is an eigenvector with eigenvalue \( \rho \), then all entries of \( x \) are nonzero and have the same sign. | Proof. The \( \rho \) -eigenspace of \( A \) is 1-dimensional, for otherwise we could find a \( \rho \) -subharmonic vector with some entry equal to zero, contradicting Lemma 8.7.1. If \( x \) is an eigenvector with eigenvalue \( \rho \), then by the previous lemma, \( \left| x\right| \) is a positive eigenvector with ... | Yes |
Theorem 8.8.1 Suppose \( A \) is a real nonnegative \( n \times n \) matrix whose underlying directed graph \( X \) is strongly connected. Then:\n\n(a) \( \rho \left( A\right) \) is a simple eigenvalue of \( A \) . If \( x \) is an eigenvector for \( \rho \), then no entries of \( x \) are zero, and all have the same s... | The first two parts of this theorem follow from the results of the previous section. We discuss part (c), but do not give a complete proof of it, since we will not need its full strength. | No |
Theorem 8.9.1 Let \( A \) be a symmetric matrix of rank \( r \) . Then there is a permutation matrix \( P \) and a principal \( r \times r \) submatrix \( M \) of \( A \) such that\n\n\[ \n{P}^{T}{AP} = \left( \begin{matrix} I \\ R \end{matrix}\right) M\left( \begin{array}{ll} I & {R}^{T} \end{array}\right) .\n\] | Proof. Since \( A \) has rank \( r \), there is a linearly independent set of \( r \) rows of \( A \) . By symmetry, the corresponding set of columns is also linearly independent. The entries of \( A \) in these rows and columns determine an \( r \times r \) principal submatrix \( M \) . Therefore, there is a permutati... | Yes |
Corollary 8.9.4 Let \( A \) be a real symmetric \( n \times n \) matrix of rank \( r \) . Then there is an \( n \times r \) matrix \( C \) of rank \( r \) such that\n\n\[ A = {CN}{C}^{T} \]\n\nwhere \( N \) is a block-diagonal \( r \times r \) matrix with \( r - {2s} \) diagonal entries equal to \( \pm 1 \), and \( s \... | Proof. We note that\n\n\[ {\beta }^{-1}\left( {y{z}^{T} + z{y}^{T}}\right) = \left( \begin{array}{ll} {\beta }^{-1}y & z \end{array}\right) \left( \begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right) {\left( \begin{array}{ll} {\beta }^{-1}y & z \end{array}\right) }^{T}. \]\n\nTherefore, if we take \( C \) to be the \( n... | Yes |
Theorem 8.10.1 Let \( A \) be a symmetric \( n \times n \) matrix over \( {GF}\left( 2\right) \) with zero diagonal and binary rank \( m \) . Then \( m \) is even and there is an \( m \times n \) matrix \( C \) of rank \( m \) such that\n\n\[ A = {CN}{C}^{T} \]\n\nwhere \( N \) is a block diagonal matrix with \( m/2 \)... | Proof. Over \( {GF}\left( 2\right) \), the diagonal entries of the matrix \( y{z}^{T} + z{y}^{T} \) are zero. Since all diagonal entries of \( A \) are zero, it follows that the algorithm implicit in the proof of Lemma 8.9.3 will express \( A \) as a sum of symmetric matrices with rank two and zero diagonals. Therefore... | Yes |
Theorem 8.11.1 A reduced graph has binary rank at most \( {2r} \) if and only if it is an induced subgraph of \( \operatorname{Sp}\left( {2r}\right) \) . | Proof. Any reduced graph \( X \) of binary rank \( {2r} \) has a vectorial representation as a spanning set of nonzero vectors in \( {GF}{\left( 2\right) }^{2r} \) . Therefore, the vertex set of \( X \) is a subset of the vertices of \( \operatorname{Sp}\left( {2r}\right) \), where two vertices are adjacent in \( X \) ... | Yes |
Theorem 8.11.2 Every graph on \( {2r} - 1 \) vertices occurs as an induced subgraph of \( \operatorname{Sp}\left( {2r}\right) \) . | Proof. We prove this by induction on \( r \) . It is true when \( r = 1 \) because a single vertex is an induced subgraph of a triangle. So suppose that \( r > 1 \) , and let \( X \) be an arbitrary graph on \( {2r} - 1 \) vertices. If \( X \) is empty, then it is straightforward to see that it is an induced subgraph o... | No |
Lemma 8.12.1 If \( X \) is a graph with diameter \( d \), then \( A\left( X\right) \) has at least \( d + 1 \) distinct eigenvalues. | Proof. We sketch the proof. Observe that the \( {uv} \) -entry of \( {\left( A + I\right) }^{r} \) is nonzero if and only if \( u \) and \( v \) are joined by a path of length at most \( r \) . Consequently, the matrices \( {\left( A + I\right) }^{r} \) for \( r = 0,\ldots, d \) form a linearly independent subset in th... | No |
Lemma 8.13.1 Let \( A \) be a real symmetric \( n \times n \) matrix and let \( B \) denote the matrix obtained by deleting the ith row and column of \( A \) . Then\n\n\[ \frac{\phi \left( {B, x}\right) }{\phi \left( {A, x}\right) } = {e}_{i}^{T}{\left( xI - A\right) }^{-1}{e}_{i} \]\n\nwhere \( {e}_{i} \) is the ith s... | Proof. From the standard determinantal formula for the inverse of a matrix we have\n\n\[ {\left( {\left( xI - A\right) }^{-1}\right) }_{ii} = \frac{\det \left( {{xI} - B}\right) }{\det \left( {{xI} - A}\right) } \]\n\nso noting that\n\n\[ {\left( {\left( xI - A\right) }^{-1}\right) }_{ii} = {e}_{i}^{T}{\left( xI - A\ri... | Yes |
Corollary 8.13.2 For any graph \( X \) we have\n\n\[ \n{\phi }^{\prime }\left( {X, x}\right) = \mathop{\sum }\limits_{{u \in V\left( X\right) }}\phi \left( {X \smallsetminus u, x}\right) \n\] | Proof. By (8.3),\n\n\[ \n\operatorname{tr}{\left( xI - A\right) }^{-1} = \mathop{\sum }\limits_{\theta }\frac{\operatorname{tr}{E}_{\theta }}{x - \theta }. \n\]\n\nBy the lemma, the left side here is equal to\n\n\[ \n\mathop{\sum }\limits_{{u \in V\left( X\right) }}\frac{\phi \left( {X \smallsetminus u, x}\right) }{\ph... | Yes |
Theorem 8.13.3 Let \( A \) be a real symmetric \( n \times n \) matrix, let \( b \) be a vector of length \( n \), and define \( \psi \left( x\right) \) to be the rational function \( {b}^{T}{\left( xI - A\right) }^{-1}b \) . Then all zeros and poles of \( \psi \) are simple, and \( {\psi }^{\prime } \) is negative eve... | Proof. By (8.3),\n\n\[ \n{b}^{T}{\left( xI - A\right) }^{-1}b = \mathop{\sum }\limits_{{\theta \in \operatorname{ev}\left( A\right) }}\frac{{b}^{T}{E}_{\theta }b}{x - \theta }. \n\]\n\n(8.4)\n\nThis implies that the poles of \( \psi \) are simple. We differentiate both sides of (8.4) to obtain\n\n\[ \n{\psi }^{\prime }... | Yes |
Theorem 9.1.1 Let \( A \) be a real symmetric \( n \times n \) matrix and let \( B \) be a principal submatrix of \( A \) with order \( m \times m \) . Then, for \( i = 1,\ldots, m \) , | Proof. We prove the result by induction on \( n \) . If \( m = n \), there is nothing to prove. Assume \( m = n - 1 \) . Then, by Lemma 8.13.1, for some \( i \) we have\n\n\[ \frac{\phi \left( {B, x}\right) }{\phi \left( {A, x}\right) } = {e}_{i}^{T}{\left( xI - A\right) }^{-1}{e}_{i} \]\n\nDenote this rational functio... | Yes |
Lemma 9.2.1 There are no Hamilton cycles in the Petersen graph P. | Proof. First note that there is a Hamilton cycle in \( P \) if and only if there is an induced \( {C}_{10} \) in \( L\left( P\right) \) . Now, \( L\left( P\right) \) has eigenvalues \( 4,2, - 1 \), and -2 with respective multiplicities \( 1,5,4 \), and 5 (see Exercise 8.9). In particular, \( {\theta }_{7}\left( {L\left... | Yes |
Lemma 9.2.2 The edges of \( {K}_{10} \) cannot be partitioned into three copies of the Petersen graph. | Proof. Let \( P \) and \( Q \) be two copies of Petersen’s graph on the same vertex set and with no edges in common. Let \( R \) be the subgraph of \( {K}_{10} \) formed by the edges not in \( P \) or \( Q \) . We show that \( R \) is bipartite.\n\nLet \( {U}_{P} \) be the eigenspace of \( A\left( P\right) \) with eige... | Yes |
Lemma 9.3.1 Let \( \pi \) be an equitable partition of the graph \( X \), with characteristic matrix \( P \), and let \( B = A\left( {X/\pi }\right) \) . Then \( {AP} = {PB} \) and \( B = \) \( {\left( {P}^{T}P\right) }^{-1}{P}^{T}{AP} \) . | Proof. We will show that for all vertices \( u \) and cells \( {C}_{j} \) we have\n\n\[ \n{\left( AP\right) }_{uj} = {\left( PB\right) }_{uj} \n\] \n\nThe \( {uj} \) -entry of \( {AP} \) is the number of neighbours of \( u \) that lie in \( {C}_{j} \) . If \( u \in {C}_{i} \), then this number is \( {b}_{ij} \) . Now, ... | Yes |
Lemma 9.3.2 Let \( X \) be a graph with adjacency matrix \( A \) and let \( \pi \) be a partition of \( V\left( X\right) \) with characteristic matrix \( P \) . Then \( \pi \) is equitable if and only if the column space of \( P \) is \( A \) -invariant. | Proof. The column space of \( P \) is \( A \) -invariant if and only if there is a matrix \( B \) such that \( {AP} = {PB} \) . If \( \pi \) is equitable, then by the previous lemma we may take \( B = A\left( {X/\pi }\right) \) . Conversely, if there is such a matrix \( B \) , then every vertex in cell \( {C}_{i} \) is... | Yes |
Theorem 9.3.3 If \( \pi \) is an equitable partition of a graph \( X \), then the characteristic polynomial of \( A\left( {X/\pi }\right) \) divides the characteristic polynomial of \( A\left( X\right) \) . | Proof. Let \( P \) be the characteristic matrix of \( \pi \) and let \( B = A\left( {X/\pi }\right) \) . If \( X \) has \( n \) vertices, then let \( Q \) be an \( n \times \left( {n - \left| \pi \right| }\right) \) matrix whose columns, together with those of \( P \), form a basis for \( {\mathbb{R}}^{n} \) . Then the... | Yes |
Lemma 9.3.4 If \( X \) is a regular graph with a perfect 1-code, then -1 is an eigenvalue of \( A\left( X\right) \) . | Proof. Let \( S \) be a perfect 1-code and consider the partition \( \pi \) of \( V\left( X\right) \) into \( S \) and its complement. If \( X \) is \( k \) -regular, then the definition of a perfect 1-code implies that \( \pi \) is equitable with quotient matrix\n\n\[ \left( \begin{matrix} 0 & k \\ 1 & k - 1 \end{matr... | Yes |
Theorem 9.4.1 Let \( X \) be a vertex-transitive graph and \( \pi \) the orbit partition of some subgroup \( G \) of \( \operatorname{Aut}\left( X\right) \) . If \( \pi \) has a singleton cell \( \{ u\} \), then every eigenvalue of \( X \) is an eigenvalue of \( X/\pi \) . | Proof. If \( f \) is a function on \( V\left( X\right) \), and \( g \in \operatorname{Aut}\left( X\right) \), then let \( {f}^{g} \) denote the function given by\n\n\[ \n{f}^{g}\left( x\right) = f\left( {x}^{g}\right) .\n\]\n\nIt is routine to show that if \( f \) is an eigenvector of \( X \) with eigenvalue \( \theta ... | Yes |
Lemma 9.4.2 We have\n\n\\[ \n\\mathop{\\sum }\\limits_{{i = 0}}^{h}{\\left( -1\\right) }^{h - i}\\left( \\begin{matrix} h \\ i \\end{matrix}\\right) \\left( \\begin{matrix} a - i \\ k \\end{matrix}\\right) = {\\left( -1\\right) }^{h}\\left( \\begin{array}{l} a - h \\ k - h \\end{array}\\right) .\n\\] | Proof. Denote the sum in the statement of the lemma by \\( f\\left( {a, h, k}\\right) \\) . Since\n\n\\[ \n\\left( \\begin{matrix} a - i \\ k \\end{matrix}\\right) = \\left( \\begin{matrix} a - i - 1 \\ k \\end{matrix}\\right) + \\left( \\begin{matrix} a - i - 1 \\ k - 1 \\end{matrix}\\right)\n\\]\n\nwe have\n\n\\[ \nf... | Yes |
Theorem 9.5.1 Let \( A \) be a real symmetric \( n \times n \) matrix and let \( R \) be an \( n \times m \) matrix such that \( {R}^{T}R = {I}_{m} \) . Set \( B \) equal to \( {R}^{T}{AR} \) and let \( {v}_{1},\ldots ,{v}_{m} \) be an orthogonal set of eigenvectors for \( B \) such that \( B{v}_{i} = {\theta }_{i}\lef... | Proof. Let \( {u}_{1},\ldots ,{u}_{n} \) be an orthogonal set of eigenvectors for \( A \) such that \( A{u}_{i} = {\theta }_{i}\left( A\right) {u}_{i} \) . Let \( {U}_{j} \) be the span of \( {u}_{1},\ldots ,{u}_{j} \) and let \( {V}_{j} \) be the span of \( {v}_{1},\ldots ,{v}_{j} \) . For any \( i \), the space \( {V... | Yes |
Lemma 9.6.1 If \( P \) is the characteristic matrix of a partition \( \pi \) of the vertices of the graph \( X \), then the eigenvalues of \( {\left( {P}^{T}P\right) }^{-1}{P}^{T}{AP} \) interlace the eigenvalues of \( A \) . If the interlacing is tight, then \( \pi \) is equitable. | Proof. The problem with \( P \) is that its columns form an orthogonal set, not an orthonormal set, but fortunately this can easily be fixed. Recall that \( {P}^{T}P \) is a diagonal matrix with positive diagonal entries, and so there is a diagonal matrix \( D \) such that \( {D}^{2} = {P}^{T}P \) . If \( R = P{D}^{-1}... | Yes |
Lemma 9.6.2 Let \( X \) be a \( k \) -regular graph on \( n \) vertices with least eigenvalue \( \tau \) . Then\n\n\[ \alpha \left( X\right) \leq \frac{n\left( {-\tau }\right) }{k - \tau } \] | Proof. The inequality follows on unpacking (9.4). If \( S \) is an independent set with size meeting this bound, then the partition with \( S \) and \( V\left( X\right) \smallsetminus S \) as its cells is equitable, and so each vertex not in \( S \) has exactly \( k\left| S\right| /\left( {n - \left| S\right| }\right) ... | Yes |
Lemma 9.6.3 Let \( X \) be a graph on \( n \) vertices and let \( A \) be a symmetric \( n \times n \) matrix such that \( {A}_{uv} = 0 \) if the vertices \( u \) and \( v \) are not adjacent. Then\n\n\[ \alpha \left( X\right) \leq \min \left\{ {n - {n}^{ + }\left( A\right), n - {n}^{ - }\left( A\right) }\right\} . \] | Proof. Let \( S \) be the subgraph of \( X \) induced by an independent set of size \( s \), and let \( B \) be the principal submatrix of \( A \) with rows and columns indexed by the vertices in \( S \) . (So \( B \) is the zero matrix.) By interlacing,\n\n\[ {\theta }_{n - s + i}\left( A\right) \leq {\theta }_{i}\lef... | Yes |
Lemma 9.8.1 A fullerene has exactly twelve 5-cycles. | Proof. Suppose \( F \) is a fullerene with \( n \) vertices, \( e \) edges, and \( f \) faces. Then \( n, e \), and \( f \) are constrained by Euler’s relation, \( n - e + f = 2 \) . Since \( F \) is cubic, \( {3n} = {2e} \) . Let \( {f}_{r} \) denote the number of faces of \( F \) with size \( r \) . Then\n\n\[
{f}_{5... | Yes |
Lemma 9.9.1 If \( X \) is a cubic planar graph with leapfrog graph \( F\left( X\right) \), then \( F\left( X\right) \) has at most half of its eigenvalues positive and at most half of its eigenvalues negative. | Proof. Let \( \pi \) be the partition whose cells are the edges of the canonical perfect matching \( M \) of \( F\left( X\right) \). Since \( X \) is cubic, two distinct cells of \( \pi \) are joined by at most one edge. The graph defined on the cells of \( \pi \) where two cells are adjacent if they are joined by an e... | Yes |
Theorem 9.9.2 If \( X \) is a cubic planar graph, then its leapfrog graph \( F\left( X\right) \) has exactly half of its eigenvalues negative. If, in addition, \( X \) has a face of length not divisible by three, then its leapfrog graph \( F\left( X\right) \) also has exactly half of its eigenvalues positive. | Proof. By the lemma, the first conclusion follows if \( {\theta }_{m + 1}\left( A\right) \neq 0 \), and the second follows if \( {\theta }_{m}\left( A\right) \neq 0 \) . Suppose to the contrary that \( {\theta }_{m + 1}\left( A\right) = 0 \) . Then by Theorem 9.5.1, there is an eigenvector \( f \) for \( A \) with eige... | Yes |
Lemma 10.1.1 Let \( X \) be an \( \left( {n, k, a, c}\right) \) strongly regular graph. Then the following are equivalent:\n\n(a) \( X \) is not connected,\n\n(b) \( c = 0 \) ,\n\n(c) \( a = k - 1 \) ,\n\n(d) \( X \) is isomorphic to \( m{K}_{k + 1} \) for some \( m > 1 \) . | Proof. Suppose that \( X \) is not connected and let \( {X}_{1} \) be a component of \( X \) . A vertex in \( {X}_{1} \) has no common neighbours with a vertex not in \( {X}_{1} \), and so \( c = 0 \) . If \( c = 0 \), then any two neighbours of a vertex \( u \in V\left( X\right) \) must be adjacent, and so \( a = k - ... | Yes |
Lemma 10.2.1 A connected regular graph with exactly three distinct eigenvalues is strongly regular. | Proof. Suppose that \( X \) is connected and regular with eigenvalues \( k,\theta \) , and \( \tau \), where \( k \) is the valency. If \( A = A\left( X\right) \), then the matrix polynomial\n\n\[ M \mathrel{\text{:=}} \frac{1}{\left( {k - \theta }\right) \left( {k - \tau }\right) }\left( {A - {\theta I}}\right) \left(... | Yes |
Lemma 10.3.1 Let \( X \) be strongly regular with parameters \( \left( {n, k, a, c}\right) \) and distinct eigenvalues \( k,\theta \), and \( \tau \) . Then \[ {m}_{\theta }{m}_{\tau } = \frac{{nk}\bar{k}}{{\left( \theta - \tau \right) }^{2}}. \] | Proof. The proof of this lemma is left as an exercise. | No |
Lemma 10.3.2 Let \( X \) be strongly regular with parameters \( \left( {n, k, a, c}\right) \) and eigenvalues \( k,\theta \), and \( \tau \) . If \( {m}_{\theta } = {m}_{\tau } \), then \( k = \left( {n - 1}\right) /2, a = \left( {n - 5}\right) /4 \) , and \( c = \left( {n - 1}\right) /4 \) . | Proof. If \( {m}_{\theta } = {m}_{\tau } \), then they both equal \( \left( {n - 1}\right) /2 \), which we denote by \( m \) . Then \( m \) is coprime to \( n \), and therefore it follows from the previous lemma that \( {m}^{2} \) divides \( k\bar{k} \) . Since \( k + \bar{k} = n - 1 \), it must be the case that \( k\b... | Yes |
Lemma 10.3.4 Let \( X \) be a strongly regular graph with \( p \) vertices, where \( p \) is prime. Then \( X \) is a conference graph. | Proof. By Lemma 10.3.1 we have\n\n\[{\left( \theta - \tau \right) }^{2} = \frac{{pk}\bar{k}}{{m}_{\theta }{m}_{\tau }}\]\n\n(10.3)\n\nIf \( X \) is not a conference graph, then \( {\left( \theta - \tau \right) }^{2} \) is a perfect square. But since \( k,\bar{k},{m}_{\theta } \), and \( {m}_{\tau } \) are all nonzero v... | Yes |
Lemma 10.3.5 Let \( X \) be a primitive strongly regular graph with an eigenvalue \( \theta \) of multiplicity \( n/2 \) . If \( k < n/2 \), then the parameters of \( X \) are\n\n\[ \left( {{\left( 2\theta + 1\right) }^{2} + 1,\theta \left( {{2\theta } + 1}\right) ,{\theta }^{2} - 1,{\theta }^{2}}\right) . | Proof. Since \( {m}_{\theta } = n - 1 - {m}_{\tau } \), we see that \( {m}_{\theta } \neq {m}_{\tau } \), and hence that \( \theta \) and \( \tau \) are integers.\n\nFirst we will show by contradiction that \( \theta \) must be the eigenvalue of multiplicity \( n/2 \) . Suppose instead that \( {m}_{\tau } = n/2 \) . Fr... | Yes |
Corollary 10.3.6 Let \( X \) be a primitive strongly regular graph with \( {2p} \) vertices where \( p \) is prime. Then the parameters of \( X \) or its complement are\n\n\[ \left( {{\left( 2\theta + 1\right) }^{2} + 1,\theta \left( {{2\theta } + 1}\right) ,{\theta }^{2} - 1,{\theta }^{2}}\right) . | Proof. By taking the complement if necessary, we may assume that \( k \leq \left( {n - 1}\right) /2 \) . The graph \( X \) cannot be a conference graph (because for a conference graph \( n = 2{m}_{\tau } + 1 \) is odd), and hence \( \theta \) and \( \tau \) are integers. Since \( {\left( \theta - \tau \right) }^{2}{m}_... | No |
Lemma 10.4.3 Let \( L \) be a Latin square of order \( n \) and let \( X \) be the graph of the corresponding \( {OA}\left( {3, n}\right) \) . Then the maximum number of vertices \( \alpha \left( X\right) \) in an independent set of \( X \) is \( n \), and the chromatic number \( \chi \left( X\right) \) of \( X \) is a... | Proof. If we identify the \( {n}^{2} \) vertices of \( X \) with the \( {n}^{2} \) cells of the Latin square \( L \), then it is clear that an independent set of \( X \) can contain at most one cell from each row of \( L \) . Therefore, \( \alpha \left( X\right) \leq n \), which immediately implies that \( \chi \left( ... | Yes |
Lemma 10.4.4 Let \( L \) be a Latin square arising from the multiplication table of the cyclic group \( G \) of order \( {2n} \) and let \( X \) be the graph of the corresponding \( {OA}\left( {3,{2n}}\right) \) . Then \( X \) has no independent sets of size \( {2n} \) . | Proof. Suppose on the contrary that \( X \) does have an independent set of \( {2n} \) vertices, described by the permutation \( \pi \) . There is a unique element \( \tau \) of order two in \( G \), and so all the remaining nonidentity elements can be paired with their inverses. It follows that the product of all the ... | Yes |
Theorem 10.4.5 An \( {OA}\left( {k, n}\right) \) is extendible if and only if its graph has chromatic number \( n \) . | Proof. Let \( X \) be the graph of an \( {OA}\left( {k, n}\right) \) . Suppose first that \( \chi \left( X\right) = n \) . Then the \( {n}^{2} \) vertices of \( X \) fall into \( n \) colour classes \( V\left( X\right) = {V}_{1} \cup \cdots \cup {V}_{n} \) . Define the \( \left( {k + 1}\right) \) st row of the orthogon... | Yes |
Lemma 10.6.1 Let \( X \) be strongly regular with eigenvalues \( k > \theta > \tau \) . Suppose that \( x \) is an eigenvector of \( {A}_{1} \) with eigenvalue \( {\sigma }_{1} \) such that \( {\mathbf{1}}^{T}x = \) 0 . If \( {Bx} = 0 \), then \( {\sigma }_{1} \in \{ \theta ,\tau \} \), and if \( {Bx} \neq 0 \), then \... | Proof. Since \( {\mathbf{1}}^{T}x = 0 \), we have\n\n\[ \left( {{A}_{1}^{2} - \left( {a - c}\right) {A}_{1} - \left( {k - c}\right) I}\right) x = - {B}^{T}{Bx} \]\n\nand since \( X \) is strongly regular with eigenvalues \( k,\theta \), and \( \tau \), we have\n\n\[ \left( {{A}_{1}^{2} - \left( {a - c}\right) {A}_{1} -... | Yes |
Theorem 10.6.3 Let \( X \) be an \( \left( {n, k, a, c}\right) \) strongly regular graph. Then \( \sigma \) is a local eigenvalue of one subconstituent of \( X \) if and only if \( a - c - \sigma \) is a local eigenvalue of the other, with equal multiplicities. | Proof. Suppose that \( {\sigma }_{1} \) is a local eigenvalue of \( {A}_{1} \) with eigenvector \( x \) . Then, since \( {\mathbf{1}}^{T}x = 0 \) ,\n\n\[ B{A}_{1} + {A}_{2}B = \left( {a - c}\right) B + {cJ} \]\n\nimplies that\n\n\[ {A}_{2}{Bx} = \left( {a - c}\right) {Bx} - {\sigma }_{1}{Bx} = \left( {a - c - {\sigma }... | Yes |
Theorem 10.6.4 The Clebsch graph is the unique strongly regular graph with parameters \( \left( {{16},5,0,2}\right) \) . | Proof. Suppose that \( X \) is a \( \left( {{16},5,0,2}\right) \) strongly regular graph, which therefore has eigenvalues \( 5,1 \), and -3 . Let \( {X}_{2} \) denote the second subcon-stituent of \( X \) . This is a cubic graph on 10 vertices, and so has an eigenvalue 3 with eigenvector 1. All its other eigenvectors a... | Yes |
Lemma 10.7.2 If \( k \geq {m}_{\theta } \), then \( \tau \) is an eigenvalue of the first subconstituent of \( X \) with multiplicity at least \( k - {m}_{\theta } \) . | Proof. Let \( U \) denote the space of functions on \( V\left( X\right) \) that sum to zero on each subconstituent of \( X \) relative to \( u \) . This space has dimension \( n - 3 \) . Let \( T \) be the space spanned by the eigenvectors of \( X \) with eigenvalue \( \tau \) that sum to zero on \( V\left( {X}_{1}\rig... | Yes |
Lemma 10.7.3 If \( k \geq {m}_{\theta } \), then\n\n\[ \n\left( {{m}_{\theta } - 1}\right) \left( {{ka} - {a}^{2} - \left( {k - {m}_{\theta }}\right) {\tau }^{2}}\right) - {\left( a + \left( k - {m}_{\theta }\right) \tau \right) }^{2} \geq 0.\n\] | Proof. We know that \( a \) is an eigenvalue of \( {A}_{1} \) with multiplicity at least one, and that \( \tau \) is an eigenvalue with multiplicity at least \( k - m \) . This leaves \( m - 1 \) eigenvalues as yet unaccounted for; we denote them by \( {\sigma }_{1},\ldots ,{\sigma }_{m - 1} \) . Then\n\n\[ \n0 = \oper... | Yes |
Lemma 10.7.4 If \( k < {m}_{\theta } \), then\n\n\[ \n\left( {{m}_{\theta } - 1}\right) \left( {{ka} - {a}^{2} - \left( {k - {m}_{\theta }}\right) {\tau }^{2}}\right) - {\left( a + \left( k - {m}_{\theta }\right) \tau \right) }^{2} > 0.\n\] | Proof. Define the polynomial \( p\left( x\right) \) by\n\n\[ \np\left( x\right) \mathrel{\text{:=}} \left( {m - 1}\right) \left( {{ka} - {a}^{2} - \left( {k - m}\right) {x}^{2}}\right) - {\left( a + \left( k - m\right) x\right) }^{2}.\n\]\n\nThen\n\n\[ \np\left( x\right) = \left( {m - 1}\right) {ka} - m{a}^{2} + {2a}\l... | Yes |
Corollary 10.7.6 There is no strongly regular graph with parameter set \( \left( {{28},9,0,4}\right) \) . | Proof. The parameter set \( \left( {{28},9,0,4}\right) \) is feasible, and a strongly regular graph with these parameters would have spectrum\n\n\[ \left\{ {9,{1}^{\left( {21}\right) }, - {5}^{\left( 6\right) }}\right\} \]\n\nBut if \( k = 9,\theta = 1 \), and \( \tau = - 5 \), then\n\n\[ {\theta }^{2}\tau - {2\theta }... | Yes |
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