Cycle neighbor polynomial of some graph operations

Abstract

The Cycle Neighbor Polynomial of a graph \(G\) is defined as, \(C N^*[G, z]=\Sigma_{k=0}^{c(G)} c_k(G) z^k\), where \(c_0(G)\) is the number of isolated vertices, \(c_1(G)\) is the number of non isolated vertices which does not belong to any cycle of \(G, c_2(G)\) is the number of bridges and \(c_k(G)\) is the number of cycles of length \(k\) in \(G\) for \(g(G) \leq k \leq c(G)\) with \(g(G)\) and \(c(G)\) are respectively the girth and circumference of \(G\). This paper deals with the cycle neighbor polynomial of some graph operations, graph modifications and that of graphs derived from the given graph.