The maximum spectral radius of P2k+1-free graphs of given size

Abstract

In this paper, we consider a Brualdi-Hoffman-Turán problem for graphs without path of given length. Denote by $C_t^{+}$ the graph obtained from a cycle $C_t$ by linking two vertices of distance two in the cycle. Recently, Li, Zhai and Shu showed that for $k\ge 3$ and a graph G of size $m\ge 16k^2{(k+2)}^2$, if G is $C_{2k+1}^{+}$-free or $C_{2k+2}^{+}$-free, then the maximum adjacency spectral radius $\rho \left(G\right)\le \frac{1}{2}(k-1+\sqrt{4m-k^2+1})$. It follows immediately that if G is $P_{2k+1}$-free of size $m\ge 16k^2{(k+2)}^2$, then $\rho \left(G\right)\le \frac{1}{2}(k-1+\sqrt{4m-k^2+1})$. However, the upper bound is not sharp. We consider the case for $P_{2k+1}$-free graphs and obtain the following result: Let $k\ge 4$ and G be a $P_{2k+1}$-free graph of size $m\ge 4{(k+1)}^4$. Then $\rho \left(G\right)\le \frac{1}{2}(k-2+\sqrt{4m-{(k-1)}^2+1})$, and equality holds if and only if $G\cong K_{k-1}\nabla (\frac{m}{k-1}-\frac{k-2}{2})K_1$.