Highly irregular graphs

Abstract

A connected graph is highly irregular if each of its vertices is adjacent only to vertices with distinct degrees. In this paper w e investigate several problems concerning the existence and enumeration of highly irregular graphs as well as their independence numbers, with particular focus on the corresponding problems for highly irregular trees.

On (2,(C1,C2)) Regular Bipolar Fuzzy Graph

“…Definition 2.12. The total degree of a vertex u ∈ V is denoted by td(u) and defined as td(u) 1 , v is the shortest path connecting u and v of length 2}. Figure 3.…”

On neighbourly irregular graphs

“…A connected graph G is said to be highly irregular if each neighbor of any vertex has different degree [1]. It is called k-neighbourhood regular if each vertex is adjacent to exactly k vertices of the same degree [5].…”

How to Define an Irregular Graph

“…These graphs do exist . Some locally irregular graphs are shown in Locally irregular graphs that are also connected have been defined and studied recently in [1], and have been referred to as highly irregular graphs . There are surprisingly many highly irregular graphs, as was pointed out in [1] .…”

“…In [1], it was proved that there exist highly irregular graphs with n vertices for every positive integer n except 3, 5, and 7 . There are only two connected graphs with three vertices, and neither of these is highly irregular since each vertex of degree 2 has two neighbors with the same degree .…”

“…In [1], it was shown that for every graph H there is a highly irregular graph G containing H as an induced subgraph . To illustrate the proof of this analogue, consider the non-highly irregular graph H in Figure 7 .…”