The booksize $\mathrm{bk}(G)$ of a graph $G$ is the largest number of triangles sharing a common edge. A classical theorem of Edwards, conjectured by Bollob\'{a}s and Erd\H{o}s, states that every $n$-vertex graph $G$ with $e(G)>e(T_{n,2})$ has booksize greater than $n/6$. Zhai and Lin [J. Graph Theory 102 (2023) 502--5...
Let $G$ be an $n$-vertex graph with $e(G)$ edges, and let $\lambda(G)$ denote the largest eigenvalue of its adjacency matrix. The booksize $\mathrm{bk} (G)$ of $G$ is defined as the largest number of triangles sharing a common edge. The main purpose of this note is to prove that if $\lambda(G)\geq\lambda(T_{n,2})$ and...
Let $G$ be a simple graph of maximum degree $d$, and let $\mu(G)$ denote the largest eigenvalue of its Laplacian matrix. For a fixed integer $k\geq 2$, Aharoni, Alon, and Berger (2016) asked whether every graph containing no induced copy of $K_{1,k}$ satisfies $\mu(G)\leq (2 - \frac{2}{k} + o(1)) d$. We answer this que...
Le-Le Liu, Bo Ning· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
