On the spectral moment of graphs with given clique number

TL;DR

This paper investigates the ordering of connected graphs with a fixed number of vertices and clique number based on spectral moments, characterizing the extremal graphs in this order for various clique numbers.

Contribution

It characterizes the first and last graphs in the spectral moment order within classes of graphs with fixed clique number, extending understanding of spectral properties in graph theory.

Findings

Identified the first graphs in the spectral order for given clique numbers.

Characterized the last graphs in the spectral order for certain clique numbers.

Provided explicit characterizations for specific cases like t=n-2 and t=n-3.

Abstract

Let $\mathscr{L}_{n,t}$ be the set of all $n$-vertex connected graphs with clique number $t$\,($2\leq t\leq n)$. For $n$-vertex connected graphs with given clique number, lexicographic ordering by spectral moments ($S$-order) is discussed in this paper. The first $\sum_{i=1}^{\lfloor\frac{n-t-1}{3}\rfloor}(n-t-3i)+1$ graphs with $3\le t\le n-4$, and the last few graphs, in the $S$-order, among $\mathscr{L}_{n,t}$ are characterized. In addition, all graphs in $\mathscr{L}_{n,n}\bigcup\mathscr{L}_{n,n-1}$ have an $S$-order; for the cases $t=n-2$ and $t=n-3$ the first three and the first seven graphs in the set $\mathscr{L}_{n,t}$ are characterized, respectively.