Measures of irregularity of graphs

Abstract

ABSTRACT.A graph is regular if every vertex is of the same degree. Otherwise, it is an irregular graph. Although there is a vast literature devoted to regular graphs, only a few papers approach the irregular ones. We have found four distinct graph invariants used to measure the irregularity of a graph. All of them are determined through either the average or the variance of the vertex degrees. Among them there is the index of the graph, a spectral parameter, which is given as a function of the maximum eigenvalue of its adjacency matrix. In this paper, we survey these invariants with highlight to their respective properties, especially those relative to extremal graphs. Finally, we determine the maximum values of those measures and characterize their extremal graphs in some special classes.

Irregularidad de grafos: Discusión, extensiones de grafos y nuevas propuestas

“…We can observe that these initiatives, when applied to some polynomial IMs, have resulted in extremal graphs with diversity equal to 2 -or, in another case, not larger than 4, for every order n, which seems contradictory in relation to the notion of irregularity, when considering its opposite -the regularity: regular graphs G have a single degree value, then ξ(G) = 1 for them. Details concerning the known extremal graph families for IMs are in [22]. i) [5] defined the variance measure, based on the variance of the vertex degree set,…”

“…The first graph, from G (12,19), is not HI, although shown as such in the reference (each 5-degree vertex is connected to a pair of 5-degree vertices). The second one, from G (14,22), built by starting with the first, is HI. The third one, from G (26,25), is an HI tree [12], [2].…”

On graph irregularity indices with particular regard to degree deviation

“…to be used as a measure of irregularity of G. It needs to be mentioned here that the degree deviation of G is actually n times the discrepancy of G [28,29]. Some mathematical properties of the degree deviation can be found in the survey [41] and papers [33,39].…”

“…This means that there is no single parameter that can be used to measure the irregularity of graphs. A relevant list of papers concerned with various irregularity measures, but hardly exhaustive, would include [1,5,8,10,11,15,16,21,23,24,33,34,39,41,[44][45][46].…”

On a Conjecture about Degree Deviation Measure of Graphs

“…Let k and n be integer numbers such that 0 ≤ k ≤ n. A graph S(n, k) is a split graph if there is a partition of its vertex set into a clique of order k and a stable set of order n − k. A complete split graph, CS(n, k), is a split graph such that each vertex of the clique is adjacent to each vertex of the stable set [5].…”