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 $G 1 K_2$ 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 energy of a graph is the sum of the absolute values of the eigenvalues of its adjacency matrix. The edge grafting operation on a graph is a kind of edge moving between two vertices of the graph. In this paper, we introduce two new edge grafting operations and show how the graph energy changes under these edge grafting operations. Let G(n) be the set of all unicyclic graphs with n vertices. ...

Irregular coloring is a proper and each vertex on graph must have different code. The color code of v where the number vertices that are adjacent to colored i. minimum k-color used in irregular called chromatic denoted by . In this paper, we discuss for bull graph, pan sun peach caveman graph.

For a graph $G = (V, E)$, a partition $pi = {V_1,$ $V_2,$ $ldots,$ $V_k}$ of the vertex set $V$ is an textit{upper domatic partition} if $V_i$ dominates $V_j$ or $V_j$ dominates $V_i$ or both for every $V_i, V_j in pi$, whenever $i neq j$. The textit{upper domatic number} $D(G)$ is the maximum order of an upper domatic partition. We study the properties of upper domatic number and propose an up...

A mixed graph means a graph containing both oriented edges and undirected edges. The nullity of the Hermitian-adjacency matrix of a mixed graph G, denoted by ηH(G), is referred to as the multiplicity of the eigenvalue zero. In this paper, for a mixed unicyclic graph G with given order and matching number, we give a formula on ηH(G), which combines the cases of undirected and oriented unicyclic ...

In this paper, we consider the edge searching problem on cycle-disjoint graphs. We first improve the running time of the algorithm to compute the vertex separation and the optimal layout of a unicyclic graph, which is given by Ellis et al. (2004), from O(n log n) to O(n). By a linear-time transformation, we can compute the edge search number of a unicyclic graph in linear time. We also propose ...

Let G be a graph with n(G) vertices and m(G) be its matching number. The nullity of G, denoted by η(G), is the multiplicity of the eigenvalue zero of adjacency matrix of G. It is well known that if G is a tree, then η(G) = n(G)− 2m(G). Guo et al. [Jiming GUO, Weigen YAN, Yeongnan YEH. On the nullity and the matching number of unicyclic graphs. Linear Alg. Appl., 2009, 431: 1293–1301] proved tha...

We define the swapping number of an arbitrary simple graph, which is related to edge reconstruction, and involves a weakening of the concept of a graph automorphism. We classify all 1-swappable trees and unicyclic graphs and prove that the expected value of the swapping number grows linearly with the order of the graph.

The Kirchhoff index of a connected graph is the sum of resistance distances between all unordered pairs of vertices in the graph. Its considerable applications are found in a variety of fields. In this paper, we determine the maximum value of Kirchhoff index among the unicyclic graphs with fixed number of vertices and maximum degree, and characterize the corresponding extremal graph.