Extensions on spectral extrema of θ1,2,5-free graphs with given size

The Brualdi-Hoffman-Turán type problem is an important category of spectral Turán problems, focusing on determining the maximum spectral radius of an $F$-free graph with m edges. This topic has received considerable attention in recent years. Let $G(m,F)$ denote the set of $F$-free graphs with m edges and no isolated vertices. Define ${\theta }_{p,q,r}$ as the theta graph, consisting of three internally disjoint paths of lengths p, q, and r, sharing the same pair of endpoints. Recently, Lu et al. (2024) [16] determined that the unique extremal graph with the maximum spectral radius in $G(m,{\theta }_{1,2,5})$ is $S_{\frac{m+6}{3},3}$ for $m\ge 38$. However, the extremal graph is well-defined only when $m\equiv 0\left(\mathrm{mod}3\right)$. In this paper, we employ a new technique to characterize the graphs with the maximum spectral radius among all ${\theta }_{1,2,5}$-free graphs for $m\ge 39$, where $m\equiv 1\left(\mathrm{mod}3\right)$ or $m\equiv 2\left(\mathrm{mod}3\right)$. Finally, we propose two conjectures that generalize Zhai et al. (2021) [31].