Search

For a (molecular) graph, the first Zagreb index M1 is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index M2 is equal to the sum of the products of the degrees of pairs of adjacent vertices. It is well-known that for connected or disconnected graphs, M2/m ≥ M1/n does not hold always. In K. C. Das (On comparing Zagreb indices of graphs, MATCH Commun. Math. Comput. Chem. 63, 433–440, 2010), it has been shown that the above relation holds for a special kind of graph. Here we continue our search for special kinds of graph for which the above relation holds.

The energy of a graph G, denoted by E(G), is the sum of the absolutevalues of all eigenvalues of G. In this paper we present some lowerand upper bounds for E(G) in terms of number of vertices, numberof edges, and determinant of the adjacency matrix. Our lower boundis better than the classical McClelland’s lower bound. In addition,Nordhaus–Gaddum type results for E(G) are established.

Let G = (V, E) be a simple graph with vertex set V (G) = (v(1,) v(2),..., v(n),} and edge set E(G). The Randie matrix R = (r(ij)) of a graph G whose vertex v(i) has degree di is defined by r(ij) = 1/root d(i)d(j) if the vertices v(i) and v(j) are adjacent and r(ij) = 0 otherwise. The Randie energy RE is the sum of absolute values of the eigenvalues of R. We provide lower and upper bounds for RE in terms of no. of vertices, maximum degree, minimum degree and the determinant of the adjacency matrix of graphs G.

As a continues study of the paper [4], in here, we first state and prove the p-Cockcroft property (or, equivalently, efficiency) for a presentation, say PE, of the semi-direct product of a free abelian monoid rank two by a finite cyclic monoid. Then, in a separate section, we present sufficient conditions on a special case for PE to be minimal whilst it is inefficient.

Let G be a connected graph of order n with Laplacian eigenvalues mu(1) >= mu(2) >= ... >= mu(n-1) > mu(n) = 0. The Laplacian energy of the graph G is defined as

Let G = (V, E) be a simple connected graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then the signless Laplacian matrix of G is Q(G) = D(G) + A(G). In this paper, we obtain some new and improved sharp upper bounds on the spectral radius q(1)(G) of the signless Laplacian matrix of a graph G. (C) 2012 Elsevier Inc. All rights reserved.