On the spectral moment of graphs with given clique number

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.