Study A Degree Sum Condition With Connectivity For Relative Length of Shortest Paths and Cycles

A graph G, p(G) denote the orders of a Shortest Path c(G) and denote the orders of a shortest cycle in G. For a graph G, we denote by d_G(y) and k(G) the degree of a vertex y in G respectively. In this paper we prove that if G is a 2-connected graph of order n such that ∑³_j=1 d_G (y_i) ≥ n + k(G) + 2 for every independent set {y₁, y₂, y₃}, then p(G) - c(G) ≤ 1.