Abstract
Energy of a graph, firstly defined by E. Hückel as the sum of absolute values of the eigenvalues of the adjacency matrix, in other words the sum of absolute values of the roots of the characteristic (spectral) polynomials, is an important sub area of graph theory. Symmetry and regularity are two important and desired properties in many areas including graphs. In many molecular graphs, we have a pointwise symmetry, that is the graph corresponding to the molecule under investigation has two identical subgraphs which are symmetrical at a vertex. Therefore, in this paper, we shall study only the vertex joining graphsIn this article we study the characteristic polynomials of the two kinds of joining graphs called splice and link graphs of some well known graph classes.
Highlights
One of the methods of studying graphs is to make use of the graph operations
The vertex joining graph at v or the splice of these two graphs is denoted by G1 ∨v G2 and obtained by identifying the vertices v of the two graphs
If two graphs G1, G2 are not labelled, the vertex at which we join these two graphs can be selected in many different ways
Summary
One of the methods of studying graphs is to make use of the graph operations. There is a large and increasing number of graph operations such as join, corona, cartesian product, union, composition, concatenation, brick product, etc. The vertex joining graph at v or the splice of these two graphs is denoted by G1 ∨v G2 and obtained by identifying the vertices v of the two graphs. We give the formula for the vertex joining graphs G ∨v G for G = Kn, Sn and Pn as follows.
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