Extensions on spectral extrema of $C_5/C_6$-free graphs with given size

Let $\mathcal{F}$ denote a set of graphs. A graph $G$ is said to be $\mathcal{F}$-free if it does not contain any element of $\mathcal{F}$ as a subgraph. The Tur\'an number is the maximum possible number of edges in an $\mathcal{F}$-free graph with $n$ vertices. It is well known that classical Tur\'an type extremal problem aims to study the Tur\'an number of fixed graphs. In 2010, Nikiforov \cite{Nik2} proposed analogously a spectral Tur\'an type problem which asks to determine the maximum spectral radius of an $\mathcal{F}$-free graph with $n$ vertices. It attracts much attention and many such problems remained elusive open even after serious attempts, and so they are considered as one of the most intriguing problems in spectral extremal graph theory. It is interesting to consider another spectral Tur\'an type problem which asks to determine the maximum spectral radius of an $\mathcal{F}$-free graph with $m$ edges. Denote by $\mathcal{G}(m,\mathcal{F})$ the set of $\mathcal{F}$-free graphs with $m$ edges having no isolated vertices. Each of the graphs among $\mathcal{G}(m,\mathcal{F})$ having the largest spectral radius is called a maximal graph. Let $\theta_{p,q,r}$ be a theta graph formed by connecting two distinct vertices with three independent paths of length $p,q$ and $r,$ respectively (length refers to the number of edges). In this paper, we firstly determine the unique maximal graph among $\mathcal{G}(m,\theta_{1,2,3})$ and $\mathcal{G}(m,\theta_{1,2,4}),$ respectively. Then we determine all the maximal graphs among $\mathcal{G}(m,C_5)$ (resp. $\mathcal{G}(m,C_6)$) excluding the book graph. These results extend some earlier results.