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