On the Spectral Radius of Bowtie-Free Graphs with Given Size

For a family of graphs $$\mathcal {H}$$ , a graph G is $$\mathcal {H}$$ -free if it does not contain a subgraph isomorphic to any graph in $$\mathcal {H}$$ . Nikiforov (Linear Algebra Appl 432:2243–2256, 2010) proposed a spectral analogue of the Turán problem which asks to determine the maximum spectral radius of an H-free graph of size m or order n. Let $$F_k$$ be the graph obtained from k intersecting triangles sharing a common vertex. For $$k=2$$ , the graph $$F_2$$ is called the bowtie. Let $$S_{n,2}$$ be the graph obtained by joining each vertex of $$K_2$$ to $$n-2$$ isolated vertices and let $$S^-_{n,2}$$ be the graph obtained from $$S_{n,2}$$ by deleting an edge incident to vertex of degree 2. Li et al. (Discrete Math 346:113680, 2023) showed that $$\rho (G)\le \frac{1+\sqrt{4m-3}}{2}$$ for any $$F_2$$ -free graph of size $$m\ge 8$$ . They also proved that the unique extremal graph is the join of $$K_2$$ with an independent set of $$\frac{m-1}{2}$$ vertices. However, this bound is only sharp for odd m. Let $$\mathcal {F}$$ be a family of graphs, where any m-edge graph G in $$\mathcal {F}$$ is obtained from a bipartite graph B(S, W) (not necessarily complete) by adding two new vertices u and $$v_0$$ , an edge $$uv_0$$ , all edges between u and S, and t edges between $$v_0$$ and S, where $$t\le |N(u)|$$ . In this paper, we determine a sharp upper bound for $$\rho (G)$$ in $$F_2$$ -free graphs that do not belong to $$\mathcal {F}$$ , given that G has $$m$$ (even) edges.