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a r t i c l e i n f o a b s t r a c t Let G be a connected k-regular graph of order n. We find a best upper bound (in terms of k) on the third largest eigenvalue that is sufficient to guarantee that G has a perfect matching when n is even, and a matching of order n − 1 when n is odd. We also examine how other eigenvalues affect the size of matchings in G.

Let G be an irregular graph on n vertices with maximum degree ∆ and diameter D. We show that ∆ − λ 1 > 1 nD , where λ 1 is the largest eigenvalue of the adjacency matrix of G. We also study the effect of adding or removing few edges on the spectral radius of a regular graph. Our graph notation is standard (see West [22]). For a graph G, we denote by λ i (G)… (More)

We study the minimum number of complete r-partite r-uniform hypergraphs needed to partition the edges of the complete r-uniform hypergraph on n vertices and we improve previous results of Alon.

Let λ 1 be the greatest eigenvalue and λ n the least eigenvalue of the adjacency matrix of a connected graph G with n vertices, m edges and diameter D. We prove that if G is nonregular, then Δ − λ 1 > nΔ − 2m n(D(nΔ − 2m) + 1) 1 n(D + 1) , where Δ is the maximum degree of G. The inequality improves previous bounds of Stevanovi´c and of Zhang. It also… (More)

In this paper, we present an elementary proof of a theorem of Serre concerning the greatest eigenvalues of k-regular graphs. We also prove an analogue of Serre's theorem regarding the least eigenvalues of k-regular graphs: given ǫ > 0, there exist a positive constant c = c(ǫ, k) and a nonnegative integer g = g(ǫ, k) such that for any k-regular graph X with… (More)

In this paper, we study the relationship between eigenvalues and the existence of certain subgraphs in regular graphs. We give a condition on an appropriate eigenvalue that guarantees a lower bound for the matching number of a t-edge-connected d-regular graph, when t ≤ d − 2. This work extends some classical results of von Baebler and Berge and more recent… (More)

For a graph G and k a real number, we consider the sum of the k-th powers of the degrees of the vertices of G. We present some general bounds on this sum for various values of k.

The chromatic number χ(G) of a graph G is the minimum number of colors in a proper coloring of the vertices of G. The biclique partition number bp(G) is the minimum number of complete bipartite subgraphs whose edges partition the edge-set of G. The Rank-Coloring Conjecture (formulated by van Nuffelen in 1976) states that χ(G) ≤ rank(A(G)), where rank(A(G))… (More)

a r t i c l e i n f o a b s t r a c t Keywords: Eigenvalues of graphs Graph spectrum Expander Edge-transitive graphs Vertex-transitive graphs Extremal graph theory Algebraically defined graphs Let q = p e , where p is a prime and e 1 is an integer. For m 1, let P and L be two copies of the (m + 1)-dimensional vector spaces over the finite field F q.… (More)

Partially answering a question of Paul Seymour, we obtain a sufficient eigenvalue condition for the existence of k edge-disjoint spanning trees in a regular graph, when k ∈ {2, 3}. More precisely, we show that if the second largest eigenvalue of a d-regular graph G is less than d − 2k−1 d+1 , then G contains at least k edge-disjoint spanning trees, when k ∈… (More)