Comparison between the non-self-centrality number and the total irregularity of graphs

Abstract

The non-self-centrality number and the total irregularity of a connected graph G are defined as N(G)=∑|εG(vi)−εG(vj)| and irrt(G)=∑|degG(vj)−degG(vi)|, with summations embracing all pairs of vertices, degG(vi) and ɛG(vi) denoting the degree and eccentricity of the vertex vi. In this paper, we show that there exists a graph G with diameter d such that irrt(G) > N(G) for any integer d ≥ 2. This gives a complete proof of Theorem 10 in Xu et al. (2018), where Xu et al. did not prove it really for d ≥ 4. Moreover, we prove that N(T) > irrt(T) for any tree T of order n ≥ 10 with diameter d≥2+2611n and maximum degree 4 avoiding degree 3, determine all trees(unicyclic graphs) and with diameter 3 and irrt(T) > N(T) and give a sufficient condition for trees with diameter 4 and irrt(T) > N(T). These partially solve Problems 26 and 27 in the above-mentioned literature.