Relations between total irregularity and non-self-centrality of graphs

For a connected graph G, with degG(vi) and ɛG(vi) denoting the degree and eccentricity of the vertex vi, the non-self-centrality number and the total irregularity of G are defined as N(G)=∑|ɛG(vj)−ɛG(vi)| and irrt(G)=∑|degG(vj)−degG(vi)|, with summations embracing all pairs of vertices. In this paper, we focus on relations between these two structural invariants. It is proved that irrt(G) > N(G) holds for almost all graphs. Some graphs are constructed for which N(G)=irrt(G). Moreover, we prove that N(T) > irrt(T) for any tree T of order n ≥ 15 with diameter d ≥ 2n/3 and maximum degree 3.