Graph Polynomials and Graph Transformations in ... - ELTE TTK TEO

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6wherec 4 =g(P 3 |1) − g(P 3 |2)(c 2 f(K 2 ) + c 3 g(K 2 |1)) 2.The original application of the generalized tree shift was the following theorem. (Unlikethe other theorems, this statement has a purely combinatorial proof.)Theorem 4.11. [2] Let W k (G) denote the number of closed walks of length k. Let T be atree and let T ′ be obtained from T by a generalized tree shift.Then W k (T ′ ) ≥ W k (T) for every k ≥ 1.Remark 4.12. Let µ 1 ≥ µ 2 ≥ · · · ≥ µ n be the eigenvalues of the adjacency matrix of thegraph G. Thenn∑W k (G) = µ k j.Thus the following theorem is an easy consequence of the previous statement.Corollary 4.13. [2] Let EE(G) = ∑ nj=1 eµ jdenote the Estrada index of the graph G. LetT be a tree and let T ′ be obtained from T by a generalized tree shift.Then EE(T ′ ) ≥ EE(T).Remark 4.14. The conjecture concerning the Estrada index prompted V. Nikiforov tostate the conjecture that the minimal value of W k (G) is attained at the path on n verticesamong the trees on n vertices. The generalized trees shift was developed to attack thisconjecture.j=15. Density Turán problemThe following problem was studied in Zoltán L. Nagy’s master thesis [8]. This part isbased on a joint work with him.Given a simple, connected graph H. Define the blown-up graph G[H] of H as follows.Replace all vertices v i ∈ V (H) by a cluster A i and we connect vertices between the clustersA i and A j (not necessarily all) if v i and v j were adjacent in H. Question: what kind ofedge densities we have to require between the clusters so that G[H] surely contains agraph isomorph with H such that the vertex corresponding to v ∈ V (H) is in the clustercorresponding to v. In what follows we call this phenomenon that H is a transversal ofG[H].In the sequel we need some technical definitions.Definition 5.1. [8, 9] A weighted blown-up graph is a blown-up graph where a nonnegativeweight w(u) is assigned to each vertex u such that the total weight of each clusteris 1. The density between two clusters isd ij =∑w(u)w(v).(u,v)∈Eu∈A i ,v∈A jDefinition 5.2. [8, 9] We define the critical edge density d crit (H) of the graph H asfollows. The number d crit (H) is the smallest number d for which it is true that wheneverthe edge density between any two clusters of G[H] is larger than d then H is surely atransversal of G[H].
7Figure 4. A blown-up graph of the diamond containing the diamond as a transversalDefinition 5.3. [6] Let x e ’s be variables assigned to each edge of a graph. The multivariatematching polynomial F is defined as follows:F(x e ,t) = ∑ M∈M( ∏ e∈Mx e )(−t) |M| ,where the summation goes over the matchings of the graph including the empty matching.Now we are ready to tell out our results. First we study the case when the graph H isa tree.Theorem 5.4. [6] Let T be a tree.(a) Assume that the edge density between the clusters A i ,A j of the blown-up graph G[H]is γ ij = 1 − r ij . Assume that F T (r e ,t) > 0 for t ∈ [0, 1]. Then G[T] surely contains T asa transversal.(b) If for the numbers γ ij = 1 − r ij , the polynomial F T (r e ,t) has a root in the interval[0, 1] then there exists weighted blown-up graph G[T] of the tree T such that the edge densitybetween the clusters A i ,A j is γ ij , till G[T] does not contain T as a transversal.Corollary 5.5. [9] Let T be a tree and µ(T) be the largest eigenvalue of the adjacencymatrix of T. Thend crit (T) = 1 − 1µ(T) 2.If H is an arbitrary graph then the following statements remain true from the abovetheorems.Theorem 5.6. [6] Let H be a simple graph. Assume that the edge density between theclusters A i ,A j of the blown-up graph G[H] is γ ij = 1 − r ij . Assume that F H (r e ,t) > 0 fort ∈ [0, 1]. Then G[H] surely contains H as a transversal.
Page 1: Summary of the Ph.D. dissertationGr
Page 4 and 5: 2The Laplacian matrix of G is L(G)
Page 6 and 7: 4Definition 4.1. [2] Let T be a tre
Page 10 and 11: 8Theorem 5.7. [6] Let H be a simple

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