Perfect Degree Support Graphs

Abstract

In a graph G(V,E), the support s(v) of a vertex v is defined as the sum of degrees of its neighbours. Let k be any positive integer. In this paper, we introduce two new concepts in graph theory, namely, k – perfect degree support graph and (k,c) – linear degree support graph. A graph G is said to be a k – perfect degree support graph (k – pds graph) if for any vertex v in G, the ratio of its support to its degree is the constant k. A graph G is called a (k, c) – linear degree support graph ((k,c) – lds graph) if, the support of any vertex is k times its degree with a constant integer c added to it. Some families of (k,c) – lds graphs and k – pds graphs have been constructed in this paper. In addition, an interesting relationship between the eigen values of the adjacency matrix of a k – pds graph with the degree sequence of G as its eigen vector has been studied.