aa r X i v : . [ m a t h . C O ] N ov ENTROPY OF SYMMETRIC GRAPHS
SEYED SAEED CHANGIZ REZAEI, AND CHRIS GODSIL
Abstract.
A graph G is called symmetric with respect to a functional F G ( P ) defined on the set of all the probability distributions on its vertexset if the distribution P ∗ maximizing F G ( P ) is uniform on V ( G ). Usingthe combinatorial definition of the entropy of a graph in terms of itsvertex packing polytope and the relationship between the graph entropyand fractional chromatic number, we prove that vertex transitive graphsare symmetric with respect to graph entropy. As the main result of thispaper, we prove that a perfect graph is symmetric with respect to graphentropy if and only if its vertices can be covered by disjoint copies of itsmaximum-size clique. Particularly, this means that a bipartite graph issymmetric with respect to graph entropy if and only if it has a perfectmatching. Introduction
The entropy of a graph is an information theoretic functional which isdefined on a graph with a probability density on its vertex set. This func-tional was originally proposed by J. K¨orner in 1973 to study the minimumnumber of codewords required for representing an information source (seeJ. K¨orner [10]).Let
V P ( G ) be the vertex packing polytope of a given graph G which isthe convex hull of the characteristic vectors of its independent sets. Let | V ( G ) | = n and P be a probability density on V ( G ). Then the entropy of G with respect to the probability density P is defined as H ( G, P ) = min a ∈ V P ( G ) n X i =1 p i log(1 /a i ) . G. Simonyi [25] showed that the maximum of the graph entropy of agiven graph over the probability density of its vertex set is equal to itsfractional chromatic number. We say a graph is symmetric with respectto graph entropy if the uniform density maximizes its entropy. We showthat vertex transitive graphs are symmetric. In this paper, we study someclasses of graphs which are symmetric with respect to graph entropy. Ourmain results are the following theorems and corollary.
Date : July 15, 2018.
Theorem.
Let G = ( V, E ) be a perfect graph and P be a probability distribu-tion on V ( G ) . Then G is symmetric with respect to graph entropy H ( G, P ) if and only if G can be covered by disjoint copies of its maximum-size cliques. As a corollary to above theorem, we have
Corollary.
Let G be a bipartite graph with parts A and B , and no isolatedvertices. Then, uniform probability distribution U over the vertices of G maximizes H ( G, P ) if and only if G has a perfect matching. A. Schrijver [23] calls a graph G a k -graph if it is k -regular and its frac-tional edge coloring number χ ′ f ( G ) is equal to k . We show that Theorem.
Let G be a k -graph with k ≥ . Then the line graph of G issymmetric with respect to graph entropy. As a corollary to this result we show that the line graph of every bridgelesscubic graph is symmetric with respect to graph entropy.J. K¨orner investigated the basic properties of the graph entropy in severalpapers from 1973 till 1992 (see J. K¨orner [10]-[16]).Let F and G be two graphs on the same vertex set V . Then the unionof graphs F and G is the graph F ∪ G with vertex set V and its edge set isthe union of the edge set of graph F and the edge set of graph G . That is V ( F ∪ G ) = V,E ( F ∪ G ) = E ( F ) ∪ E ( G ) . The most important property of the entropy of a graph is that it is sub-additive with respect to the union of graphs, that is H ( F ∪ G, P ) ≤ H ( F, P ) + H ( G, P ) . This leads to the application of graph entropy for graph covering problemas well as the problem of perfect hashing.The graph covering problem can be described as follows. Given a graph G and a family of graphs G where each graph G i ∈ G has the same vertexset as G , we want to cover the edge set of G with the minimum number ofgraphs from G . Using the sub-additivity of graph entropy one can obtainlower bounds on this number.Graph entropy was used in a paper by Fredman and Koml´os for theminimum number of perfect hash functions of a given range that hash all k -element subsets of a set of a given size (see Fredman and Koml´os [ ? ]).2. Preliminaries
Entropy of Graphs.
Let G be a graph on vertex set V ( G ) = { , · · · , n } , and P = ( p , · · · , p n ) be a probability density on V ( G ). The vertex packing polytope of a graph G , i.e., V P ( G ), is the convex hull of thecharacteristic vectors of its independent sets. NTROPY OF SYMMETRIC GRAPHS 3
The entropy of G with respect to P , i.e., H ( G, P ), is then defined as H ( G, P ) = min a ∈ V P ( G ) n X i =1 p i log(1 /a i ) . Remark 2.1.
Note that the function P ki =1 p i log a i in the definition of thegraph entropy is a convex function and tends to infinity at the boundaryof the non-negative orthant and tends monotonically to −∞ along the raysfrom the origin.The main properties of graph entropy are monotonicity , sub-additivity ,and additivity under vertex substitution . Monotonicity is formulated in thefollowing lemma (see G. Simonyi [25]).2.1. Lemma. (J. K¨orner).
Let F be a spanning subgraph of a graph G . Thenfor any probability density P we have H ( F, P ) ≤ H ( G, P ) . The sub-additivity was first recognized by K¨orner in [12] and he provedthe following lemma.2.2.
Lemma. (J. K¨orner).
Let F and G be two graphs on the same vertexset V and F ∪ G denote the graph on V with edge set E ( F ) ∪ E ( G ) . Forany fixed probability density P we have H ( F ∪ G, P ) ≤ H ( F, P ) + H ( G, P ) . The notion of substitution is defined as follows. Let F and G be twovertex disjoint graphs and v be a vertex of G . By substituting F for v we mean deleting v and joining every vertex of F to those vertices of G which have been adjacent with v . We will denote the resulting graph G v ← F .We extend this concept also to distributions. If we are given a probabilitydistribution P on V ( G ) and a probability distribution Q on V ( F ) then by P v ← Q we denote the distribution on V ( G v ← F ) given by P v ← Q ( x ) = P ( x )if x ∈ V ( G ) \ v and P v ← Q ( x ) = P ( x ) Q ( x ) if x ∈ V ( F ). This operation isillustrated in Figure 1.Now we state the following lemma whose proof can be found in J. K¨orner,et. al. [14].2.3. Lemma. (J. K¨orner, G. Simonyi, and Zs. Tuza).
Let F and G betwo vertex disjoint graphs, v a vertex of G , while P and Q are probabilitydistributions on V ( G ) and V ( F ) , respectively. Then we have H ( G v ← F , P v ← Q ) = H ( G, P ) + P ( v ) H ( F, Q ) . Notice that the entropy of an empty graph (a graph with no edges) isalways zero (regardless of the distribution on its vertices). Noting this fact,we have the following corollary as a consequence of Lemma 2.3.
CHANGIZ REZAEI, AND GODSIL u u u u u (a) A 5-cycle G . v v v (b) Atriangle F . v v v u u u u (c) The graph G u ←− F Figure 1
Corollary.
Let the connected components of the graph G be the sub-graphs G i and P be a probability distribution on V ( G ) . If x ∈ V ( G i ) , set P i ( x ) = P ( x ) ( P ( V ( G i ))) − , x ∈ V ( G i ) . Then H ( G, P ) = X i P ( V ( G i )) H ( G i , P i ) . Now we look at entropy of some graphs which are also mentioned in G.Simonyi [24] and [25] . The first one is the complete graph.2.5.
Lemma.
For K n , the complete graph on n vertices, one has H ( K n , P ) = H ( P ) . And the next one is the complete multipartite graph. Let G = K m ,m , ··· ,m k denote a complete k -partite graph with parts of size m , m , · · · , m k . Then we have the following lemma. NTROPY OF SYMMETRIC GRAPHS 5
Lemma.
Let G = K m ,m , ··· ,m k . Given a distribution P on V ( G ) let Q be the distribution on S ( G ) , the set of maximal independent sets of G , givenby Q ( J ) = P x ∈ J P ( x ) for each J ∈ S ( G ) . Then H ( G, P ) = H ( K k , Q ) . A special case of the above Lemma is the entropy of a complete bipartitegraph with equal probability measure on its stable sets equal to 1. Now, let G be a bipartite graph with color classes A and B . For a subset D ⊆ A ,let N ( D ) denotes the the set of neighbors of D in B , that is a subset of thevertices in B which are adjacent to a vertex in A .Given a distribution P on V ( G ) we have P ( D ) = X i ∈ D p i , ∀ D ⊆ V ( G ) , Furthermore, defining the binary entropy as h ( x ) := − x log x − (1 − x ) log(1 − x ) , ≤ x ≤ , J. K¨orner and K. Marton proved the following theorem in [15].2.7.
Theorem. (J. K¨orner and K. Marton).
Let G be a bipartite graph withno isolated vertices and P be a probability distribution on its vertex set. If P ( D ) P ( A ) ≤ P ( N ( D )) P ( B ) , for all subsets D of A , then H ( G, P ) = h ( P ( A )) . And if P ( D ) P ( A ) > P ( N ( D )) P ( B ) , then there exists a partition of A = D ∪ · · · ∪ D k and a partition of B = U ∪ · · · ∪ U k such that H ( G, P ) = k X i =1 P ( D i ∪ U i ) h (cid:18) P ( D i ) P ( D i ∪ U i ) (cid:19) . Minimum Entropy Colouring.
In this section, we explain minimumentropy colouring of the vertex set of a probabilistic graph (
G, P ) which waspreviously studied by N. Alon and A. Orlitsky [1].Let X be a random variable distributed over a countable set V and π bea partition of V , i.e., π = { C , · · · , C k } and V = ∪ ki =1 C i . Then π induces aprobability distribution on its cells, that is p ( C i ) = X v ∈ C i p ( v ) , ∀ i ∈ { , · · · , k } . Therefore, the cells of π have a well-defined entropy as follows: CHANGIZ REZAEI, AND GODSIL H ( π ) = k X i =1 p ( C i ) log 1 p ( C i ) , If we consider V as the vertex set of a probabilistic graph ( G, P ) and π as a partitioning of the vertices of G into colour classes, then H ( π ) is theentropy of a proper colouring of V ( G ).The chromatic entropy of a probabilistic graph ( G, P ) is defined as H χ ( G, P ) := min { H ( π ) : π is a colouring of G } , i.e. the lowest entropy of any colouring of G . Consider a 5-cycle with twodifferent probability distributions over its vertices, i.e., uniform distributionand another one given by p = 0 . p = p = p = 0 .
2, and p = 0 .
1. Inboth of them we require three colours. In the first one, a colour is assignedto a single vertex and each of the other two colours are assigned to twovertices. Therefore, the chromatic entropy of the first probabilistic 5-cycle,i.e., H (0 . , . , .
2) is approximately equal to 1.52.For the second probabilistic 5-cycle, the chromatic entropy H (0 . , . , . { , } , { , } , and { } .The following lemmas were proved in N. Alon and A. Orlitsky [1] and J.Cardinal et. al. [3].2.8. Lemma. (N. Alon and A. Orlitsky).
Let U be the uniform distribu-tion over the vertices V ( G ) of a probabilistic graph ( G, U ) and α ( G ) be theindependence number of the graph G . Then, H χ ( G, U ) ≥ log | V ( G ) | α ( G ) . Let α ( G, P ) denote the maximum weight P ( S ) of an independent set S of a probabilistic graph ( G, P ). Then we have the following lemma.2.9.
Lemma. (J. Cardinal et. al.).
For every probabilistic graph ( G, P ) , wehave − log α ( G, P ) ≤ H ( G, P ) ≤ H χ ( G, P ) ≤ log χ ( G ) . It may seem that non-uniform distribution decreases chromatic entropy H χ ( G, P ), but the following example shows that this is not true. Let us con-sider 7-star with deg ( v ) = 7 and deg ( v i ) = 1 for i ∈ { , · · · , } . If p ( v ) =0 . p ( v i ) = for i ∈ { , · · · , } , then H χ ( G, P ) = H (0 . , .
5) = 1, whileif p ( v i ) = for i ∈ { , · · · , } , then H χ ( G, P ) = H ( , ) ≤ H (0 . , .
5) = 1.Let G , · · · , G n be graphs with vertex sets V , · · · , V n . The OR product of G , · · · , G n is the graph W ni =1 G i whose vertex set is V n and where twodistinct vertices ( v , · · · , v n ) and ( v ′ , · · · , v ′ n ) are adjacent if for some i ∈{ , · · · , n } such that v i = v ′ i , v i is adjacent to v ′ i in G i . The n -fold ORproduct of G with itself is denoted by G W n . NTROPY OF SYMMETRIC GRAPHS 7
N. Alon and A. Orlitsky [1] proved the following lemma which relateschromatic entropy to graph entropy.2.10.
Lemma. (N. Alon and A. Orlitsky).lim n →∞ n H χ ( G W n , P ( n ) ) = H ( G, P ) . Karush Kuhn Tucker (KKT) Conditions.
Karush Kuhn Tucker(KKT) optimality conditions in convex optimization are one of our tools inthis paper. Thus, we explain these conditions briefly in this section. For acomprehensive explanation see Stephen Boyd, and Lieven Vanderberghe [2]section 5.5.3. Let f ( x ) : R n → R ,f i ( x ) : R n → R , i = 1 , · · · , m,h i ( x ) : R n → R , i = 1 , · · · , p. be convex differentiable functions. We consider the following convex opti-mization problem, minimize f ( x )(1) subject to f i ( x ) ≤ , i = 1 , · · · , m,h i ( x ) = 0 , i = 1 , · · · , p. Letting λ i and ν i be the Lagrange multipliers corresponding to constraints f i ( x ) and h i ( x ), respectively, we define the Lagrangian L : R n × R m × R p → R as L ( x , λ, ν ) = f ( x ) + m X i =1 λ i f i ( x ) + p X i =1 ν i h i ( x ) . Then x ∗ is an optimal solution to (1) if and only if there exist Lagrangemultipliers λ ∗ and ν ∗ such that the following conditions hold, f i ( x ∗ ) ≤ , i = 1 , · · · , m, (2) h i ( x ∗ ) = 0 , i = 1 , · · · , p,λ ∗ i ≥ , i = 1 , · · · , m,λ ∗ i f i ( x ∗ ) = 0 , i = 1 , · · · , m ∇ f ( x ∗ ) + m X i =1 λ ∗ i ∇ f i ( x ∗ ) + p X i =1 ν ∗ i ∇ h i ( x ∗ ) = 0 . The conditions (2) above are called
Karush Kuhn Tucker (KKT) optimalityconditions.3.
Graph Entropy and Fractional Chromatic Number
In this section we investigate the relation between the entropy of a graphand its fractional chromatic number which was already established by G.Simonyi [25]. First we recall that the fractional chromatic number of a
CHANGIZ REZAEI, AND GODSIL graph G denoted by χ f ( G ) is the minimum sum of nonnegative weights onthe independent sets of G such that for any vertex the sum of the weightson the independent sets of G containing that vertex is at least one (see C.Godsil and G. Royle [9] sections 7.1 to 7.5).3.1. Lemma. (G. Simonyi).
For a graph G and probability density P on itsvertices with fractional chromatic number χ f ( G ) , we have max P H ( G, P ) = log χ f ( G ) . Proof.
Note that for every graph G we have (cid:16) χ f ( G ) , · · · , χ f ( G ) (cid:17) ∈ V P ( G ).Thus for every probability density P , we have H ( G, P ) ≤ log χ f ( G ) . Now, we show that graph G has an induced subgraph G ′ with χ f ( G ′ ) = χ f ( G ) = χ f such that if y ∈ V P ( G ′ ) and y ≥ χ f , then y = χ f .Suppose the above statement does not hold for graph G . Consider all y ∈ V P ( G ) such that y ≥ χ f ( G ) . Note that there is not any y ∈ V P ( G ) such that y > χ f ( G ) , because then we have a fractional colouring with value strictly less than χ f ( G ). Thus for every y ≥ χ f ( G ) there is some v ∈ V ( G ) such that y v = χ f ( G ) . For such a fixed y , letΩ y = (cid:26) v ∈ V ( G ) : y v > χ f ( G ) (cid:27) . Let y ∗ be one of those y ’s with | Ω y | of maximum size. Let G ′ = G [ V ( G ) \ Ω y ∗ ] . From our the definition of G ′ and fractional chromatic number, we haveeither χ f ( G ′ ) < χ f ( G ) . or ∃ y ∈ V P (cid:0) G ′ (cid:1) , such that y ≥ χ f and y = χ f . Suppose χ f ( G ′ ) < χ f ( G ) . Therefore z = χ f ( G ′ ) ∈ V P ( G ′ ) NTROPY OF SYMMETRIC GRAPHS 9 and consequently z > χ f ( G ) . Without loss of generality assume that V ( G ) \ V ( G ′ ) = { , · · · , | V ( G ) \ V ( G ′ ) |} . Set ǫ := 12 (cid:18) min v ∈ Ω y ∗ y v − χ f ( G ) (cid:19) > , z ∗ = (cid:16) T | V ( G ) \ V ( G ′ ) | , z T (cid:17) T ∈ V P ( G ) . Then (1 − ǫ ) y ∗ + ǫ z ∗ ∈ V P ( G ) , which contradicts the maximality assumption of Ω y ∗ . Thus, we have χ f ( G ′ ) = χ f ( G ) . Now we prove that if y ∈ V P ( G ′ ) and y ≥ χ f , then y = χ f .Suppose z ′ be a point in V P ( G ′ ) such that z ′ ≥ χ f but z ′ = χ f . Set y ′ = (cid:16) T | V ( G ) \ V ( G ′ ) | , z ′ T (cid:17) T ∈ V P ( G ) . Then using the ǫ > − ǫ ) y ∗ + ǫ y ′ ∈ V P ( G ) , which contradicts the maximality assumption of Ω y ∗ .Now, by I.Csisz´ar et. al. [6], there exists a probability density P ′ on V P ( G ′ ) such that H ( G ′ , P ′ ) = log χ f . Extending P ′ to a probability distri-bution P as(3) p i = (cid:26) p ′ i , i ∈ V ( G ) , , i ∈ V ( G ) \ V ( G ′ ) . the lemma is proved. Indeed, suppose that H ( G, P ) < H ( G ′ , P ′ ) and let¯ y ∈ V P ( G ) be a point in V P ( G ) which gives H ( G, P ). Let ¯ y V P ( G ′ ) be therestriction of ¯ y in V P ( G ′ ). Then there exists z ∈ V P ( G ′ ) such that z ≥ ¯ y V P ( G ′ ) . This contradicts the fact that H (cid:0) G ′ , P ′ (cid:1) = log χ f . Symmetric Graphs A symmetric graph with respect to graph entropy is a graph whose entropyis maximized with uniform probability distribution over its vertex set. Inthis section, we characterize different classes of graphs which are symmetricwith respect to graph entropy. Particularly, we consider symmetric bipartitegraphs, symmetric perfect graphs, and symmetric line graphs.First, using chromatic entropy introduced in section 2.2, we show thatevery vertex transitive graph is symmetric with respect to graph entropy.4.1. Theorem.
Let G be a vertex transitive graph. Then the uniformdistribution over vertices of G maximizes H ( G, P ) . That is H ( G, U ) =log χ f ( G ) . Proof.
First note that for a vertex transitive graph G , we have χ f ( G ) = | V ( G ) | α ( G ) , and the n -fold OR product G W n of a vertex transitive graph G isalso vertex transitive. Now from Lemma 2.8, Lemma 2.9 , and Lemma 3.1,we have(4) H (cid:16) G W n , U (cid:17) ≤ log χ f (cid:16) G W n (cid:17) ≤ H χ (cid:16) G W n , U (cid:17) , From N. Alon and A. Orltsky [1] and D. Ullman and E. Scheinerman [26],we have H (cid:0) G W n , U (cid:1) = nH ( G, U ), χ f (cid:0) G W n (cid:1) = χ f ( G ) n , and log χ f ( G ) =lim n →∞ n log χ (cid:0) G W n (cid:1) . Hence, applying Lemma 2.10 to equation (4) andusing squeezing theorem, we get(5) H ( G, U ) = log χ f ( G ) = lim n →∞ n log χ (cid:16) G W n (cid:17) = lim n →∞ n H χ (cid:16) G W n , U (cid:17) . The following example shows that the converse of the above theorem isnot true. Consider G = C ∪ C , with vertex sets V ( C ) = { v , v , v , v } and V ( C ) = { v , v , v , v , v , v } , and parts A = { v , v , v , v , v } , B = { v , v , v , v , v } . Clearly, G is not a vertex transitive graph, however, us-ing Theorem 2.7, one can see that the uniform distribution U = (cid:0) , · · · , (cid:1) gives the maximum graph entropy which is 1. Remark 4.1.
Note that the probability distribution which maximizesthe graph entropy is not unique. Consider C with vertex set V ( C ) = { v , v , v , v } with parts A = { v , v } and B = { v , v } . Using Theorem2.7, probability distributions P = ( , , , ) and P = ( , , , ) give themaximum graph entropy which is 1.4.1. Symmetric Perfect Graphs.
Let G = ( V, E ) be a graph. Recallthat the fractional vertex packing polytope of G ,i.e, F V P ( G ) is defined as F V P ( G ) := { x ∈ R | V | + : X v ∈ K x v ≤ } . NTROPY OF SYMMETRIC GRAPHS 11
Note that for every graph G , we have V P ( G ) ⊆ F V P ( G ) . The following theorem was previously proved in V. Chv´atal [5] and D. R.Fulkerson [8].4.2.
Theorem.
A graph G is perfect if and only if V P ( G ) = F V P ( G ) . The following theorem which is called weak perfect graph theorem is usefulin the following discussion. This theorem was proved by Lov´asz in [18] and[19] and is follows.4.3.
Theorem.
A graph G is perfect if and only if its complement is per-fect. Now, we prove the following theorem which is a generalization of ourbipartite symmetric graphs with respect to graph entropy.4.4.
Theorem.
Let G = ( V, E ) be a perfect graph and P be a probabilitydistribution on V ( G ) . Then G is symmetric with respect to graph entropy H ( G, P ) if and only if G can be covered by disjoint copies of its maximum-size cliques. Proof.
Suppose G is covered by its maximum-sized cliques, say Q , · · · , Q m .That is V ( G ) = V ( Q ) ˙ ∪ · · · ˙ ∪ V ( Q m ) and | V ( Q i ) | = ω ( G ) , ∀ i ∈ [ m ].Now, consider graph T which is the disjoint union of the subgraphs in-duced by V ( Q i ) ∀ i ∈ [ m ]. That T = ˙ S mi =1 G [ V ( Q i )]. Noting that T is adisconnected graph with m components, using Corollary 2.4 we have H ( T, P ) = X i P ( Q i ) H ( Q i , P i ) . Now, having V ( T ) = V ( G ) and E ( T ) ⊆ E ( G ), we get H ( T, P ) ≤ H ( G, P ) for every distribution P . Using Lemma 3.1, this implies(6) H ( T, P ) = X i P ( Q i ) H ( Q i , P i ) ≤ log χ f ( G ) , ∀ P, Noting that G is a perfect graph, the fact that complete graphs are sym-metric with respect to graph entropy, χ f ( Q i ) = χ f ( G ) = ω ( G ) = χ ( G ) , ∀ i ∈ [ m ], and (6), we conclude that uniform distribution maximizes H ( G, P ).Now, suppose that G is symmetric with respect to graph entropy. Weprove that G can be covered by its maximum-sized cliques. Suppose this isnot true. We show that G is not symmetric with respect to H ( G, P ).Denoting the minimum clique cover number of G by γ ( G ) and the maxi-mum independent set number of G by α ( G ), from perfection of G and weakperfect theorem, we get γ ( G ) = α ( G ). Then, using this fact, our assumptionimplies that G has an independent set S with | S | > | V ( G ) | ω ( G ) . We define a vector x such that x v = | S || V | if v ∈ S and x v = − | S || V | ω − if v ∈ V ( G ) \ S . Then, we can see that x ∈ F V P ( G ) = V P ( G ). Let t := | S || V | .Then, noting that t > ω , H ( G, U ) ≤ − | V | X v ∈ V log x v = − | V | X v ∈ S log x v + X v ∈ V \ S x v = − | V | (cid:18) | S | log α + ( | V | − | S | ) log 1 − αω − (cid:19) = − t log t − (1 − t ) log 1 − tω − − t log t − ( ω − (cid:18) − tω − − tω − (cid:19) < log ω ( G ) . Note that we have γ ( G ) = α ( G ) . The above theorem about symmetric perfect graphs is specialized to bi-partite graphs in the following corollary. We give a separate proof for thefollowing corollary, in S. S. C. Rezaei [4], using Hall’s theorem, K¨onig’stheorem (see D. West [28] section 3.1) and Theorem 2.7.4.5.
Corollary. (Symmetric Bipartite Graphs).
Let G be a bipartite graphwith parts A and B , and no isolated vertices. Then, uniform probabilitydistribution U over the vertices of G maximizes H ( G, P ) if and only if G has a perfect matching. We also have the following corollary.4.6.
Corollary.
Let G be a connected regular line graph without any isolatedvertices with valency k > . Then if G is covered by copies of its disjointmaximum-size cliques, then G is symmetric with respect to H ( G, P ) . Proof.
Let G = L ( H ) for some graph H . Then either H is bipartite or regular. If H is bipartite, then G is perfect (see D. West [28] sections 7.1 and 8.1) andbecause of Theorem 4.4 we are done. So suppose that H is not bipartite.Then each clique of size k in G corresponds to a vertex v in V ( H ) and theedges incident to v in H and vice versa. That is because any such cliques in G contains a triangle and there is only one way extending that triangle tothe whole clique which corresponds to edges incident with the correspondingvertex in H . This implies that the covering cliques in G give an independent NTROPY OF SYMMETRIC GRAPHS 13 set in H which is also a vertex cover in H . Hence H is a bipartite graphand hence G is perfect. Then due to Theorem 4.4 the theorem is proved.Now, considering that finding the clique number of a perfect graph canbe computed in polynomial time and using the weak perfect graph theorem,we conclude that one can decide in polynomial time whether a perfect graphis symmetric with respect to graph entropy.4.2. Symmetric Line Graphs.
Let G be a line graph of some graph G , i.e, G = L ( G ). Let | V ( G ) | = n and | E ( G ) | = m . Note thatevery matching in G corresponds to an independent set in G and everyindependent set in G corresponds to a matching in G . Furthermore, thefractional edge-colouring number of G , i.e., χ ′ f ( G ) is equal to the fractionalchromatic number of G , i.e., χ f ( G ). Thus χ ′ f ( G ) = χ f ( G ) . Moreover, note that the vertex packing polytope
V P ( G ) of G is the match-ing polytope M P ( G ) of G (see L. Lov´asz and M. D. Plummer [21] chapter12). That is V P ( G ) = M P ( G ) . These facts motivate us to study line graphs which are symmetric withrespect to graph entropy.We recall that a vector x ∈ R m + is in the matching polytope M P ( G ) ofthe graph G if and only if it satisfies (see A. Schrijver [23]). x e ≥ ∀ e ∈ E ( G ) ,x ( δ ( v )) ≤ ∀ v ∈ V ( G ) , (7) x ( E [ U ]) ≤ ⌊ | U |⌋ , ∀ U ⊆ V ( G ) with | U | odd.Let M denote the family of all matchings in G , and for every matching M ∈ M let the charactersitic vector b M ∈ R m + be as(8) ( b M ) e = (cid:26) , e ∈ M, , e / ∈ M. Then the fractional edge-colouring number χ ′ f ( G ) of G is defined as χ ′ f ( G ) := min { X M ∈M λ M | λ ∈ R M + , X M ∈M λ M b M = } . If we restrict λ M to be an integer, then the above definition give rise to theedge colouring number of G , i.e., χ ′ ( G ). Thus χ ′ f ( G ) ≤ χ ′ ( G ) . As an example considering G to be the Petersen graph, we have χ ′ f ( G ) = χ ′ ( G ) = 3 . The following theorem which was proved by Edmonds, gives a character-ization of the fractional edge-colouring number χ ′ f ( G ) of a graph G (seeA. Schrijver [23] section 28.6).4.7. Theorem.
Let ∆( G ) denote the maximum degree of G . Then thefractional edge-colouring number of G is obtained as χ ′ f ( G ) = max { ∆( G ) , max U ⊆ V, | U |≥ | E ( U ) |⌊ | U |⌋ } . Following A. Schrijver [23] we call a graph G a k -graph if it is k -regularand its fractional edge coloring number χ ′ f ( H ) is equal to k . The followingcorollary characterizes a k -graph (see A. Schrijver [23] section 28.6).4.8. Corollary.
Let G = ( V , E ) be a k -regular graph. Then χ ′ f ( G ) = k if and only if | δ ( U ) | ≥ k for each odd subset U of V . The following theorem introduces a class of symmetric line graphs withrespect to graph entropy. The main tool in the proof of the following theoremis Karush-Kuhn-Tucker (KKT) optimality conditions in convex optimizationexplained in section 2.3.4.9.
Theorem.
Let G be a k -graph with k ≥ . Then the line graph G = L ( G ) is symmetric with respect to graph entropy. Proof.
From our discussion above we have H ( G , P ) = min x ∈ MP ( G ) X e ∈ E ( G ) p e log 1 x e , Let λ v , γ U ≥ x ( δ ( v )) ≤ x ( E [ U ]) ≤ ⌊ | U |⌋ in the description of the matching poly-tope M P ( G ) in (7) for all v ∈ V ( G ) and for all U ⊆ V ( G ) with | U | odd,and | U | ≥
3, repectively. From our discussion in Remark 2.1, the Lagrangemulitipliers corresponding to inequalities x e ≥ g ( x ) = − X e ∈ E ( G ) p e log x e , Then the Lagrangian of g ( x ) is L ( x , λ, γ ) = − X e ∈ E ( G ) p e log x e + X e = { u,v } ( λ u + λ v ) ( x e − X e ∈ E ( G ) X U ⊆ V,U ∋ e, | U | odd , | U |≥ γ U x e − X U ⊆ V, | U | odd , | U |≥ ⌊ | U |⌋ , (9) NTROPY OF SYMMETRIC GRAPHS 15
Using KKT conditions (see S. Boyd, and L. Vanderberghe [2] section 5.5.3),the vector x ∗ minimizes g ( x ) if and only if it satisfies ∂L∂x ∗ e = 0 , → − p e x ∗ e + ( λ u + λ v ) + X U ⊆ V,U ∋ e, | U | odd , | U |≥ γ U = 0 for e = { u, v } . (10)Fix the probability density to be uniform over the edges of G , that is p e = 1 m , ∀ e ∈ E ( G ) , Note that the vector k is a feasible point in the matching polytope M P ( G ).Now, one can verify that specializing the variables as x ∗ = k ,γ U = 0 ∀ U ⊆ V, | U | odd , | U | ≥ λ u = λ v = k m ∀ e = { u, v } . satisfies the equations (10). Thus H ( G , U ) = log k. Then using Lemma 3.1 and the assumption χ f ( G ) = k the theorem isproved.It is well known that cubic graphs has a lot of interesting structure. Forexample, it can be checked that every edge in a bridgeless cubic graph isin a perfect matching. Now we have the following interesting statement forevery cubic bridgeless graph.4.10. Corollary.
The line graph of every cubic bridgeless graph G =( V , E ) is symmetric with respect to graph entropy. Proof.
We may assume that G is connected. Let U ⊆ V and let U ⊆ U consist of vertices v such that δ ( v ) ∩ δ ( U ) = ∅ . Then using handshakinglemma for G [ U ], we have3 | U | + 3 | U \ U | − | δ ( U ) | = 2 | E ( G [ U ]) | . And consequently, 3 | U | = | δ ( U ) | mod 2 , Assuming | U | is odd and noting that G is bridgeless, we have δ ( U ) ≥ . Then, considering Corollary 4.8, the corollary is proved. v v v v v v Figure 2.
A bridgeless cubic graph. v v v v v v v v v v Figure 3.
A cubic one-edge connected graph.Figure 2 shows a bridgeless cubic graph which is not edge transitive andits edges are not covered by disjoint copies of stars and triangles. Thusthe line graph of the shown graph in Figure 2 is neither vertex transitivenor covered by disjoint copies of its maximum size cliques. However, it issymmetric with respect to graph entropy by Corollary 4.10.Figure 3 shows a cubic graph with a bridge. The fractional edge chromaticnumber of this graph is 3 . . . . > . Open Questions
In this section we point out some of the problems that are worth consid-ering for future research.5.1.
Symmetric Graphs.
We defined symmetric graphs with respect tograph entropy in this paper. As the main result of this paper, we charac-terized symmetric perfect graphs. Furthermore, we proved that there are
NTROPY OF SYMMETRIC GRAPHS 17 some other classes of graphs such as vertex transitive graphs and line graphof bridgeless cubic graphs which are symmetric with respect to graph en-tropy. From what discussed above and the symmetric graphs considered inthis paper, the next natural class of graphs to consider is strongly regulargraphs . It is worth investigating if strongly regular graphs are symmetricwith respect to graph entropy. For the study of the structural properties ofstrongly regular graphs see Godsil and Royle[9] chapter 10. Furthermore, itis worth noting that finding the main properties of symmetric graphs withrespect to graph entropy is another interesting open problem.5.2.
Normal Graph Conjecture.
Let G be a graph. A set A of subsetsof V ( G ) is a covering , if every vertex of G is contained in an element of A .We say that graph G is Normal if there exists two coverings C and S such that every element C of C is a clique and every element S of S is anindependent set and the intersection of any element of C and any elementof S is nonempty, i.e., C ∩ S = ∅ , ∀ C ∈ C , S ∈ S . Recall from the sub-additivity of Graph Entropy, we have(11) H ( P ) ≤ H ( G, P ) + H ( G, P ) . A probabilistic graph (
G, P ) is weakly splitting if there exists a nowherezero probability distribution P on its vertex set which makes inequality (11)equality. The following lemma was proved in J. K¨orner et. al. [14].5.1. Lemma. (J. K¨orner, G. Simonyi, and Zs. Tuza)
A graph G is weaklysplitting if and only if it is normal. It is known that every perfect graph is also a normal graph (see J. K¨orner[11]). The following conjecture was proposed in C. De Simone and J. K¨orner[7].5.2.
Conjecture. (Normal Graph Conjecture).
A graph is hereditarily nor-mal if and only if the graph nor its complement contains C or C as aninduced subgraph. A circulant C kn is a graph with vertex set { , · · · , n } , and two vertices i = j are adjacent if and only if i − j ≡ k mod n. We assume k ≥ n ≥ k +1) to avoid cases where C kn is an independentset or a clique. A. K. Wagler [27] proved the Normal Graph Conjecture forcirculants C kn .One direction for future research is investigating the Normal Graph Con-jecture for general circulants and Cayley graphs. Graph Entropy and Graph Homomorphism.
Given a probabilis-tic graph (
G, P ), K. Marton in K. Marton [22] introduced a functional λ ( G, P ) which is analogous to Lov´asz’s bound ϑ ( G ) on Shannon capacityof graphs. Similar to ϑ ( G ), the probabilistic functional λ ( G, P ) is based onthe concept of orthonormal representation of a graph which is recalled here.Let U = { u i : i ∈ V ( G ) } be a set of unit vectors of a common dimension d such that u Ti u j = 0 if i = j and { i, j } / ∈ E ( G ) . Let c be a unit vector of dimension d . Then, the system ( U, c ) is called anorthonormal representation of the graph G with handle c .Letting T ( G ) denote the set of all orthonormal representations for graph G with a handle c , Lov´asz [20] defined ϑ ( G ) = min ( U, c ) ∈ T ( G ) max i ∈ V ( G ) u i , c ) . Then it is shown in Lov´asz [20] that zero error Shannon capacity C ( G ) canbe bounded above by ϑ ( G ) as C ( G ) ≤ log ϑ ( G ) . Let P denote the probability distribution over the vertices of G , and ǫ > capacity of the graph relative to P is C ( G, P ) = lim ǫ → lim sup n →∞ n log α (cid:16) G ( P, ǫ ) (cid:17) . A probabilistic version of ϑ ( G ) denoted by λ ( G, P ) is defined in K. Marton[22] as λ ( G, P ) := min ( U, c ) ∈ T ( G ) X i ∈ V ( G ) P i log 1( u i , c ) . K. Marton [22] showed that5.3.
Theorem. (K. Marton)
The capacity of a probabilistic graph ( G, P ) isbounded above by λ ( G, P ) , i.e., C ( G, P ) ≤ λ ( G, P ) . The following theorem was proved in K. Marton [22] which relates λ ( G, P )to H ( G, P ).5.4.
Theorem. (K. Marton)
For any probabilistic graph ( G, P ) , λ (cid:0) G, P (cid:1) ≤ H ( G, P ) . Furthermore, equality holds if and only if G is perfect. K. Marton [22] also related λ ( G, P ) to ϑ ( G ) by showing(12) max P λ ( G, P ) = log ϑ ( G ) . It is worth mentioning that ϑ ( G ) can be defined in terms of graph homo-morphisms as follows. NTROPY OF SYMMETRIC GRAPHS 19
Let d ∈ N and α <
0. Then we define S ( d, α ) to be an infinite graphwhose vertices are unit vectors in R d . Two vertices u and v are adjacent ifand only if uv T = α . Then(13) ϑ (cid:0) G (cid:1) = min (cid:26) − α : G → S ( d, α ) , α < (cid:27) . Thus, noting (12) and (13) and the above discussion, investigating the rela-tionship between graph homomorphism and graph entropy which may leadto investigating the relationship between graph homomorphism and graphcovering problem seems interesting.
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Graph Theory in Paris Trends in Mathematics
Seyed Saeed Changiz Rezaei, Chris Godsil, Department of Combinatoricsand Optimization, University of Waterloo, Waterloo, Canada
E-mail address : { sschangi,cgodsil }}