Extremal spectral radius of nonregular graphs with a fixed odd maximum degree

Abstract

For integers $n\ge 3$ and $2\le \Delta \le n-1$, let ${\lambda }_1(n,\Delta )$ be the maximum adjacency spectral radius among all connected nonregular graphs of order $n$ and maximum degree $\Delta$. Liu conjectured that for each fixed integer $\Delta \ge 3$, \[ \lim_{n\to\infty}n^2\bigl(Δ-λ_1(n,Δ)\bigr)= \begin{cases} (Δ-1)π^2/4,&\text{if $\Delta$ is odd};\\ (Δ-2)π^2/2,&\text{if $\Delta$ is even}. \end{cases} \] He proved the case $\Delta =3$ and $\Delta =4$. We prove the conjecture when $\Delta \ge 5$ is odd.