Some Basic Properties of the Second Multiplicative Zagreb Eccentricity Index

The second multiplicative Zagreb eccentricity index E 2 (G) of a simple connected graph G is expressed as the product of the weights G(a) G(b) over all edges ab of G, where 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.