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

It is well-known that the Brualdi-Hoffman-Tur\'an-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 \( m \geqslant \frac{9}{4}k^6 + 6k^5 + 46k^4 + 56k^3 + 196k^2 \) and \( k \geqslant 3 \), if \( G \) is \( F_{2k + 2} \)-free, then \( \lambda(G) \leqslant \frac{k - 1 + \sqrt{4m - k^2 + 1}}{2} \). 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 \( m \geqslant \frac{9}{4}k^6 + 6k^5 + 46k^4 + 56k^3 + 196k^2 \) with \( k \geqslant 3 \), if \( G \) is \( F_{k,3} \)-free and has \( m \) edges, then \( \lambda(G) \leqslant \frac{k - 1 + \sqrt{4m - k^2 + 1}}{2} \). 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 \( q \geqslant p \geqslant 3 \) and \( p + q \geqslant 2k + 1 \). A further research problem is also proposed.