Abstract
Following Estrada [The Structure of Complex Networks, Oxford University Press, Oxford, UK, 2011], we use the word "network" interchangeably with "graph" in this paper. Several classes of particular nonregular graphs have been introduced in literature; for example, the classes of n-vertex connected antiregular graphs, highly irregular graphs, totally segregated graphs, maximally irregular graphs and connected stepwise irregular graphs. We denote these classes by AR(n), HIRn, TSn, MIRn, SIRn, respectively. The main purpose of the present study is to identify the graph classes A, B is an element of {AR(n), HIRn, TSn, MIRn, SIRn} for which one of the following properties holds: (i) A boolean AND B = empty set (ii) A subset of B (iii) B subset of A, where A not equal B. Additionally, a new class of certain nonregular graphs, containing the class of the highly irregular graphs as a subclass, is proposed. We refer these new type of nonregular graphs as the almost highly irregular graphs and report their basic properties.