Large graphs can be found in a wide array of scientific fields ranging from sociology and biology to scientometrics and computer science. Their analysis is by no means a trivial task due to their sheer size and complex structure. Such structure encompasses features so diverse as diameter shrinking, power law degree distribution and self similarity, edge interdependence, and communities. When th...

The (n, k)-arrangement graph A(n, k) is a graph with all the k-permutations of an n-element set as vertices where two k-permutations are adjacent if they agree in exactly k − 1 positions. We introduce a cyclic decomposition for k-permutations and show that this gives rise to a very fine equitable partition of A(n, k). This equitable partition can be employed to compute the complete set of eigen...

We give some new bounds for the clique and independence numbers of a graph in terms of its eigenvalues. In particular we prove the following results. Let G be a graph of order n, average degree d, independence number α (G) , and clique number ω (G) . (i) If μn is the smallest eigenvalue of G, then ω (G) ≥ 1 + dn (n− d) (d− μn) . Equality holds if and only if G is a complete regular ω-partite gr...

The spectral radius ρ G of a graph G is the largest eigenvalue of its adjacency matrix. Let λ G be the smallest eigenvalue of G. In this paper, we have described the K3,3-minor free graphs and showed that A let G be a simple graph with order n ≥ 7. If G has no K3,3-minor, then ρ G ≤ 1 √3n − 8. B LetG be a simple connected graph with order n ≥ 3. IfG has noK3,3-minor, then λ G ≥ −√2n − 4, where ...

Scale free graphs can be found very often as models of real networks and are characterized by a power law degree distribution, that is, for a constant γ ≥ 1 the number of vertices of degree d is proportional to d−γ . Experimental studies show that the eigenvalue distribution also follows a power law for the highest eigenvalues. Hence it has been conjectured that the power law of the degrees det...

Suppose a graph G have n vertices, m edges, and t triangles. Letting λn (G) be the largest eigenvalue of the Laplacian of G and μn (G) be the smallest eigenvalue of its adjacency matrix, we prove that λn (G) ≥ 2m2 − 3nt m (n2 − 2m) , μn (G) ≤ 3n3t− 4m3 nm (n2 − 2m) , with equality if and only if G is a regular complete multipartite graph. Moreover, if G is Kr+1-free, then λn (G) ≥ 2mn (r − 1) (...

Let G be a simple, undirected graph and Gw be the positive weighted graph obtained from G by giving weights to its edges using the positive weight function w. A weighted graph Gw is said to be nonsingular if its adjacency matrix A(Gw) is nonsingular. Let G denote the class of connected, unweighted, bipartite graphs G with a unique perfect matching M such that G/M (the graph obtained by contract...