On The Roots And Stability Of Vertex Connectivity Polynomial Of Graphs

Abstract

The connectedness property of vertices in a graph is not in general preserved after the removal of geodesics connecting them. Infact, the adjacent and nonadjacent vertices in a graph may sometimes differ in terms of ”closeness” and this motivated the authors to generalise the adjacency property by introducing the concept of closely-connected in a graph. More details on closely-connected vertices are discussed in [Priya and Anil Kumar]. The study of vertex-connected vertices in [Priya and Anil Kumar, 2021a] which does not alter graph connectivity triggered the need to define vertex connectivity polynomial discussed in [Priya and Anil Kumar, 2021b] for simple finite connected graphs to explicitly reveal the number of vertex pairs that disconnect a graph.  The introduction of vertex connectivity polynomial in [Priya and Anil Kumar, 2021b] resulted in the  study of nature of roots as well as the stability properties of the same for various graph classes. This paper mainly deals with results about the nature of roots, stability and schur stability of the vertex connectivity polynomial.