Some Basic Properties of the Second Multiplicative Zagreb Eccentricity Index

Abstract

‎The second multiplicative Zagreb eccentricity index $E^{*}_{2} ({G})$ of a simple connected‎ graph $G$ is expressed as the product of the weights‎ $\varepsilon_{G}(a)\varepsilon_{G}(b)$ over all edges $ab$ of $G$‎, where $\varepsilon_{G}(a)$ stands for the‎ eccentricity of the vertex $a$ in $G$‎. ‎In this‎ paper‎, ‎some extremal problems on the $E^{*}_{2}$ index over some special graph classes including‎ trees‎, ‎unicyclic graphs and bicyclic graphs are examined‎, ‎and‎ ‎the corresponding extremal graphs are characterized‎. ‎Besides‎, ‎the relationships between this vertex-eccentricity-based graph invariant and some well-known parameters of graphs and existing graph invariants such as the number of vertices‎, ‎number of edges‎, ‎minimum vertex degree‎, ‎maximum vertex degree‎, ‎eccentric connectivity index‎, ‎connective eccentricity index‎, ‎first multiplicative Zagreb eccentricity index and second multiplicative Zagreb index are investigated‎.