Let G be a graph with integral edge weights. A function d : V (G) → Zp is called a nowhere 0 mod p domination function if each v ∈ V satisfies ( d(v) + ∑ u∈N(v) w(u, v)d(u) ) 6= 0 mod p, where w(u, v) denotes the weight of the edge (u, v) and N(v) is the neighborhood of v. The subset of vertices with d(v) 6= 0 is called a nowhere 0 mod p dominating set. It is known that every graph has a nowher...

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.

The purpose of this paper is to obtain a necessary and sufficient condition for the tensor product of two or more graphs to be connected, bipartite or eulerian. Also, we present a characterization of the duplicate graph G⊕K2 to be unicyclic. Finally, the girth and the formula for computing the number of triangles in the tensor product of graphs are worked out.

The detour index of a connected graph is defined as the sum of detour distances between all unordered pairs of vertices. We determine the n-vertex unicyclic graphs whose vertices on its unique cycle all have degree at least three with the first, the second and the third smallest and largest detour indices respectively for n ≥ 7.

Abstract. For a connected graph, the distance spectral radius is the largest eigenvalue of its distance matrix. Let U1 2k be the graph obtained from C3 by attaching a path of length n− 3 at one vertex. Let U2 2k be the graph obtained from C3 by attaching a pendant edge together with k − 2 paths of length 2 at the same vertex. In this paper, it is proved that U1 2k (resp., U 2 2k) is the unique ...

The signless Laplacian separator of a graph is defined as the difference between the largest eigenvalue and the second largest eigenvalue of the associated signless Laplacian matrix. In this paper, we determine the maximum signless Laplacian separators of unicyclic, bicyclic and tricyclic graphs with given order.

A graph with no isolated vertices is edge critical with respect to total restrained domination if for any non-edge e of G, the total restrained domination number of G+ e is less than the total restrained domination number of G. We call these graphs γtr-edge critical. In this paper, we characterize all γtr-edge critical unicyclic graphs.

The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. In this paper, we determined the extremal (maximal and minimal) unicyclic and bicyclic graphs with respect to Harary index. 2010 Mathematics Subject Classification: 05C90

In this paper, we determine the unique graph whose least signless Laplacian eigenvalue attains the minimum among all non-bipartite unicyclic graphs of order n with maximum degree Δ and among all non-bipartite connected graphs of order n with maximum degree Δ, respectively.

The Wiener index of a connected graph is defined as the sum of distances between all unordered pairs of its vertices. We determine the minimum Wiener indices of trees and unicyclic graphs with given number of vertices and matching number, respectively. The extremal graphs are characterized.