Comparison and Extremal Results on Three Eccentricity-based Invariants of Graphs

The first and second Zagreb eccentricity indices of graph G are defined as: E1(G)= n-ary sumation vi is an element of V(G)epsilon G(vi)2, E2(G)= n-ary sumation vivj is an element of E(G)epsilon G(vi)epsilon G(vj) where epsilon(G)(upsilon(i)) denotes the eccentricity of vertex upsilon(i) in G. The eccentric complexity C-ec(G) of G is the number of different eccentricities of vertices in G. In this paper we present some results on the comparison between E1(G)n and E2(G)m for any connected graphs G of order n with m edges, including general graphs and the graphs with given C-ec. Moreover, a Nordhaus-Gaddum type result C-ec(G) + C-ec(G) is determined with extremal graphs at which the upper and lower bounds are attained respectively.