Let k be a natural number and let G be a graph with at least k vertices. A.E. Brouwer conjectured that the sum of the k largest Laplacian eigenvalues of G is at most e(G) (k choose 2), where e(G) is the number of edges of G. We prove this conjecture for k = 2. We also show that if G is a tree,...

We give some necessary conditions for a graph to be 3-chromatic in terms of the spectrum of the adjacency matrix. For all known distance-regular graphs it is determined whether they are 3-chromatic. A start is made with the classification of 3-chromatic distance-regular graphs, and it is shown...

The energy of a graph T is the sum of the absolute values of the eigenvalues of the adjacency matrix of T. Seidel switching is an operation on the edge set of T. In some special cases Seidel switching does not change the spectrum, and therefore the energy. Here we investigate when Seidel...

Consider a graph Ґ on n vertices with adjacency matrix A and degree sequence (d1, . . . , dn). A universal adjacency matrix of Ґ is any matrix in Span {A,D, I, J} with a nonzero coefficient for A, where D = diag (d1, . . . , dn) and I and J are the n × n identity and all-ones matrix,...

We look for the maximum order m(r) of the adjacency matrix A of a graph G with a fixed rank r, provided A has no repeated rows or all-zero row. Akbari, Cameron and Khosrovshahi conjecture that m(r) = 2(r 2)/2−2 if r is even, and m(r) = 5 • 2(r−3)/2 − 2 if r is odd. We prove the...

We give sufficient conditions for existence of a perfect matching in a graph in terms of the eigenvalues of the Laplacian matrix. We also show that a distance-regular graph of degree k is k-edge-connected