Maximum degree and spectral radius of graphs in terms of size

Research on the relationship of the (signless Laplacian) spectral radius of a graph with its structure properties is an important research project in spectral graph theory. Denote by $ρ(G)$ and $q(G)$ the spectral radius and the signless Laplacian spectral radius of a graph $G$, respectively. Let $k\ge 0$ be a fixed integer and $G$ be a graph of size $m$ which is large enough. We show that if $ρ(G)\ge\sqrt{m-k}$, then $C_4\subseteq G$ or $K_{1,m-k}\subseteq G$. Furthermore, we prove that if $q(G)\ge m-k$, then $K_{1,m-k}\subseteq G$. Both these two results extend some known results.

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