In this paper characterizations of connected unicyclic and bicyclic graphs in terms of the degree sequence, as well as the graphs in these classes minimal with respect to the degree distance are given. © 2007 Elsevier B.V. All rights reserved.

We determine the minimum hyper-Wiener index of unicyclic graphs with given number of vertices and matching number, and characterize the extremal graphs. Mathematics Subject Classification (2010): 05C12, 05C35, 05C90.

Let U(n, k) be the set of non-bipartite unicyclic graphs with n vertices and k pendant vertices, where n ≥ 4. In this paper, the unique graph with the minimal least eigenvalue of the signless Laplacian among all graphs in U(n, k) is determined. Furthermore, it is proved that the minimal least eigenvalue of the signless Laplacian is an increasing function in k. Let Un denote the set of non-bipar...

A k-partition Π = {S1,S2, . . . ,Sk} of V (G) is resolving if for every two distinct vertices u and v of a connected graph G, there is a set Si in Π so that the minimum distance between u and a vertex of Si is different from the minimum distance between v and a vertex of Si. A resolving partition Π is said to be connected if each subgraph < Si > induced by Si (1 ≤ i ≤ k) is connected in G. In t...

Let G be a simple connected graph with pendant vertex set ∂V and nonpendant vertex set V0. The signless Laplacian matrix of G is denoted by Q(G). The signless Dirichlet eigenvalue is a real number λ such that there exists a function f ̸= 0 on V (G) such that Q(G)f(u) = λf(u) for u ∈ V0 and f(u) = 0 for u ∈ ∂V . The signless Dirichlet spectral radius λ(G) is the largest signless Dirichlet eigenva...

In this paper, we determine upper and lower bounds for the number of independent sets in a unicyclic graph in terms of its order. This gives an upper bound for the number of independent sets in a connected graph which contains at least one cycle. We also determine the upper bound for the number of independent sets in a unicyclic graph in terms of order and girth. In each case we characterize th...

Let U(n, k) be the set of non-bipartite unicyclic graphs with n vertices and k pendant vertices, where n ≥ 4. In this paper, the unique graph with the minimal least eigenvalue of the signless Laplacian among all graphs in U(n, k) is determined. Furthermore, it is proved that the minimal least eigenvalue of the signless Laplacian is an increasing function in k. Let Un denote the set of non-bipar...

Let G be a (p, q)-graph with edge domination number γ′ and edge domatic number d′. In this paper we characterize connected graphs for which γ′ = p/2 and graphs for which γ′ + d′ = q + 1. We also characterize trees and unicyclic graphs for which γ′ = bp/2c and γ′ = q −∆′, where ∆′ denotes the maximum degree of an edge in G.

Let G be a graph with n vertices and ν(G) be the matching number of G. The inertia of a graph G, In(G) = (n+, n−, n0) is an integer triple specifying the numbers of positive, negative and zero eigenvalues of the adjacency matrix A(G), respectively. Let η(G) = n0 denote the nullity of G (the multiplicity of the eigenvalue zero of G). It is well known that if G is a tree, then η(G) = n− 2ν(G). Gu...