It was conjectured that for each simple graph G = (V , E) with n = |V (G)| vertices and m = |E(G)| edges, it holdsM2(G)/m ≥ M1(G)/n, whereM1 andM2 are the first and second Zagreb indices. Hansen and Vukičević proved that it is true for all chemical graphs and does not hold in general. Also the conjecture was proved for all trees, unicyclic graphs, and all bicyclic graphs except one class. In th...

Let G be a simple graph possessing n vertices and m edges. Let di be the degree of the i-th vertex of G , i = 1, . . . , n . The first Zagreb index M1 is the sum of d 2 i over all vertices of G . The second Zagreb index M2 is the sum di dj over pairs of adjacent vertices of G . In this paper we search for graph for which M1/n = M2/m , and show how numerous such graphs can be constructed. In add...

The conjecture Σuv V(G) dG(u)2 / n(G) ≤ Σuvv E(G) dG(u)dG(v) / m(G) that compares normalized Zagreb indices attracted recently a lot of attention1-9. In this paper we analyze analogous statement in which degree dG(u) of vertex u is replaced by its eccentricity δG(u) in which way we define novel first and second Zagreb eccentricity indices. We show that Σuv V(G) εG(u)2 / n(G) ≥ Σuvv E(G) εG(u)εG...