On Eccentricity Version of Zagreb Coindices

The eccentric connectivity coindex has recently been introduced (Hua and Miao, 2019) as the total eccentricity sum of all pairs of non-adjacent vertices in a graph. Considering the total eccentricity product of non-adjacent vertex pairs, we introduce here another invariant of connected graphs called the second Zagreb eccentricity coindex. We study some mathematical properties of the eccentric connectivity coindex and second Zagreb eccentricity coindex. We also determine the extremal values of the second Zagreb eccentricity coindex over some specific families of graphs such as trees, unicyclic graphs, connected graphs, and connected bipartite graphs and describe the extremal graphs. Moreover, we compare the second Zagreb eccentricity coindex with the eccentric connectivity coindex and give directions for further studies.