some basic properties of the second multiplicative zagreb eccentricity index

The second multiplicative zagreb eccentricity index e2 (g)of a simple connected graph g is expressed as the product ofthe 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 e2 index over some specialgraph classes including trees, unicyclic graphs and bicyclicgraphs are examined, and the corresponding extremal graphsare characterized. besides, the relationships between thisvertex-eccentricity-based graph invariant and some well-knownparameters of graphs and existing graph invariants such as thenumber of vertices, number of edges, minimum vertex degree,maximum vertex degree, eccentric connectivity index, connectiveeccentricity index, first multiplicative zagreb eccentricityindex and second multiplicative zagreb index are investigated.