Spectral extrema of graphs with fixed size: forbidden a fan graph, friendship graph or theta graph

It is well-known that the Brualdi-Hoffman-Turán-type problem inquiries about the maximum spectral radius $\lambda (G)$ of an $F$-free graph $G$ with $m$ edges. Let ${\theta }_{1,p,q}$ denote the theta graph, which is constructed by connecting two vertices with 3 internally disjoint paths of lengths 1, $p$, and $q$ respectively. Let $F_k$ be the fan graph, that is, the join of a $K_1$ and a path $P_{k-1}$. Let $F_{k,3}$ be the friendship graph, obtained by having $k$ triangles share a common vertex. In this paper, we utilize the $k$-core method and spectral techniques to address some spectral extrema of graphs with a fixed number of edges. Firstly, we demonstrate that for $m⩾\frac{9}{4}{k}^6+6k^5+46k^4+56k^3+196k^2$ and $k⩾3$, if $G$ is $F_{2k+2}$-free, then $\lambda (G)⩽\frac{k-1+\sqrt{4m-k^2+1}}{2}$. Equality holds if and only if $G\cong K_k\vee (\frac{m}{k}-\frac{k-1}{2})K_1$. This validates a conjecture by Yu, Li, and Peng [Discrete Math. 348 (2025) 114391] and refines a recent result by Li, Zhai, and Shu [European J. Combin. 120 (2024) 103966]. Secondly, we show that for $m⩾\frac{9}{4}{k}^6+6k^5+46k^4+56k^3+196k^2$ with $k⩾3$, if $G$ is $F_{k,3}$-free and has $m$ edges, then $\lambda (G)⩽\frac{k-1+\sqrt{4m-k^2+1}}{2}$. Equality holds precisely when $G\cong K_k\vee (\frac{m}{k}-\frac{k-1}{2})K_1$. This confirms a conjecture put forward by Li, Lu, and Peng [Discrete Math. 346(2023)113680]. Finally, we identify the ${\theta }_{1,p,q}$-free graph with $m$ edges that possesses the largest spectral radius, where $q⩾p⩾3$ and $p+q⩾2k+1$. A further research problem is also proposed.