Mathematical Properties of Inverse Sum Index Eccentric Coindices of Graphs

Abstract

Essential and widely studied topological indices, including the well-known Zagreb indices (M1 and M2), and the newly proposed Inverse Sum Indeg Eccentricity Index (ξISI), To ensure the contributions of all edges within a graph are effectively considered. By emphasizing on the total eccentricity of non-adjacent vertices, Hua et al. introduced the eccentric connectivity coindex (ξc). Inspired by their contributions, we  introduce the inverse sum indeg eccentric coindex (ξISI), which is defined as the ratio of the product of  the eccentricities to the sum of the eccentricities for all isolated pair of vertex in a connected graph. This  study primarily aims to establish various bounds for ξISI in finite simple graphs and derives the values of  the proposed indices for two specific graph constructions. Additionally, we present a comprehensive set of  relationships for ξISI using several graph products.