For a (molecular) graph, the first Zagreb index M1 is equal to the sum of squares of the vertex degrees, and the second Zagreb index M2 is equal to the sum of products of degrees of pairs of adjacent vertices. In this paper, we study the Zagreb indices of n-vertex connected graphs with k cut vertices, the upper bound for M1and M2-values of n-vertex connected graphs with k cut vertices are deter...

Let G be a connected graph of order n. The remoteness of G, denoted by ρ, is the maximum average distance from a vertex to all other vertices. Let [Formula: see text], [Formula: see text] and [Formula: see text] be the distance, distance Laplacian and distance signless Laplacian eigenvalues of G, respectively. In this paper, we give lower bounds on [Formula: see text], [Formula: see text], [For...

We show that a k-edge-connected graph on n vertices has at least n(k/2)n−1 spanning trees. This bound is tight if k is even and the extremal graph is the n-cycle with edge-multiplicities k/2. For k odd, however, there is a lower bound cn−1 k where ck > k/2. Specifically, c3 > 1.77 and c5 > 2.75. Not surprisingly, c3 is smaller than the corresponding number for 4-edge-connected graphs. Examples ...

Notation. Given a graph, hypergraph Gn, . . . , the upper index always denotes the number of vertices, e(G), v(G) and χ(G) denote the number of edges, vertices and the chromatic number of G respectively. Given a family L of graphs, hypergraphs, ex(n,L) denotes the maximum number of edges (hyperedges) a graph (hypergraph)Gn of order n can have without containing subgraphs (subhypergraphs) from L...

For any prime power q ≥ 3, we consider two infinite series of bipartite q–regular edge–transitive graphs of orders 2q and 2q which are induced subgraphs of regular generalized 4–gon and 6–gon, respectively. We compare these two series with two families of graphs,H3(p) andH5(p), p is a prime, constructed recently by Wenger ([26]), which are new examples of extremal graphs without 6– and 10–cycle...

In this note we study some properties of the spectrum of a connected graph G with four different eigenvalues, and (spectrally maximum) diameter three. When G is regular, this is the case, for instance, when G is a distance-regular graph. © 2006 Elsevier B.V. All rights reserved.