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

Abstract: It is well-known that Brualdi-Hoffman-Tur\'an-type problem asks what is the maximum spectral radius $\lambda(G)$ of an $F$-free graph $G$ with $m$ edges? It can be viewed as a spectral characterization on the existence of the subgraph $F$ in $G.$ A nice contribution on the above problem is due to Nikiforov (2002), which states that $\lambda(G)\leqslant \sqrt{2m(1-1/r)}$ 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 3 internally disjoint paths of lengths $1,p,q$, respectively. Let $F_k$ be the fan graph, i.e., 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 use $k$-core method and spectral techniques to resolve some spectral extrema of graphs with fixed size. Firstly, we show that, for $k\geqslant3$ and $m\geqslant \frac{9}{4}k^6+6k^5+46k^4+56k^3+196k^2$, 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 $G \cong K_k\vee(\frac{m}{k}-\frac{k-1}{2})K_1.$ Secondly, we show that, for $k\geqslant 3$ and $m\geqslant \frac{9}{4}k^6+6k^5+46k^4+56k^3+196k^2$, if $G$ is $F_{k,3}$-free of size $m$, 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 confirms a conjecture proposed by Li, Lu and Peng [Discrete Math. 346(2023)113680]. Finally, we identify the $\theta_{1,p,q}$-free graph of size $m$ having the largest spectral radius, where $q\geqslant p\geqslant 3$ and $p+q\geqslant 2k+1.$ Some further research problems are also proposed.