Spectral Extrema of Graphs With Fixed Size: Forbidden a Fan Graph, a Friendship Graph, or a Theta Graph

It is well-known that Brualdi-Hoffman-Turán-type problem inquiries about the maximum spectral radius $\lambda \left(G\right)$ of an $F$-free graph $G$ with $m$ edges. This can be regarded as a spectral characterization of the existence of the subgraph $F$ within $G$. A significant contribution to this problem was made by Nikiforov (2002). He proved that $\lambda \left(G\right)⩽\sqrt{2m\left(1-1∕r\right)}$ for every $K_{r+1}$-free graph of size $m$. Let ${\theta }_{1,p,q}$ be the theta graph, which is obtained by connecting two vertices with three internally disjoint paths of lengths $1,p,q$, respectively. Let $F_k$ be the fan graph, that is, the join of a $K_1$ and a path $P_{k-1}$, and let $F_{k,3}$ be the friendship graph obtained from $k$ triangles by sharing a common vertex. In this paper, we utilize the $k$-core method and spectral techniques to address some spectral extrema of graphs with fixed size. Firstly, we show that, for $m⩾\frac{9}{4}{k}^6+6k^5+46k^4+56k^3+196k^2$ with $k⩾3$, if $G$ is $F_{2k+2}$-free, then $\lambda \left(G\right)⩽\frac{k-1+\sqrt{4m-k^2+1}}{2}$. Equality holds if and only if $G\cong K_k\vee \left(\frac{m}{k}-\frac{k-1}{2}\right)K_1$. This confirms a conjecture of Yu et al. and improves a recent result of Li et al. 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 of size $m$, then $\lambda \left(G\right)⩽\frac{k-1+\sqrt{4m-k^2+1}}{2}$. Equality holds if and only if $G\cong K_k\vee \left(\frac{m}{k}-\frac{k-1}{2}\right)K_1$. This confirms a conjecture proposed by Li et al. Finally, we identify the ${\theta }_{1,p,q}$-free graph of size $m$ having the largest spectral radius, where $q⩾p⩾3$ and $p+q⩾2k+1$. A further research problem is also proposed.