On the Irregularity Characterization of Mean Graphs
Tamás Réti, István Barányi · Acta Polytechnica Hungarica · 2021
A connected non-regular graph G with n vertices and m edges is called a mean graph, if there exists a p ≥ 2 integer for which p=[G]=2 m/n holds.The topological index p=p(G) is called the centrality parameter of graph G.It is obvious that, if G is a mean graph, then its centrality parameter p(G) is a uniquely defined positive integer.Mean graphs represent a particular subset of connected non-regular graphs.In this note, by presenting relevant examples, some structural irregularity properties of mean graphs are studied and characterized.Comparing the degree deviations S(G) and S(H) of mean graphs G and H having equal centrality parameter p(G)=p(H) it is proved that if the only difference in the corresponding degree sequences of G and H is that the number of vertices of degree p is different, then S(G)=S(H).The smallest mean graph is the 4-vertex unicyclic graph having a degree sequence (3, 2, 2, 1).This graph is isomorphic to the 4-vertex antiregular graph A4, for which S(A4)=2 holds.Using comparative tests on preselected connected graphs it has been shown that the degree deviation S(G) is poorly suited for discriminating among non-regular graphs.