On the inverse Collatz-Sinogowitz irregularity problem

Abstract

The Collatz-Sinogowitz irregularity index is the oldest known numerical measure of graph irregularity. For a simple and connected graph GG of order nn and size mm, it is defined as CS(G)=λ1−2m/n,\hspace{0.1em}\text{CS}\hspace{0.1em}\left(G)={\lambda }_{1}-2m\hspace{0.1em}\text{/}\hspace{0.1em}n, where λ1{\lambda }_{1} is the largest eigenvalue of the adjacency matrix of GG, and 2m/n2m\hspace{0.1em}\text{/}\hspace{0.1em}n is the average vertex degree of GG. Here, the Collatz-Sinogowitz inverse irregularity problem is studied. For every integer i≥0i\ge 0, it is shown that there exists a graph GG such that CS(G)=i\hspace{0.1em}\text{CS}\hspace{0.1em}\left(G)=i. Also, for every interval Ii=(i,i+1){I}_{i}=\left(i,i+1), it is shown that there are infinitely many graphs whose Collatz-Sinogowitz irregularity lies in Ii{I}_{i}.

Keywords

• irregularity measure
• collatz-sinogowitz irregularity
• inverse irregularity problem
• 05c50