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 $G$ contains a triangle, then $\mathrm{bk} (G)\geq \lambda(G) - n/3$. To obtain this goal, we prove the sharper quantitative estimate $\mathrm{bk} (G)\geq\lambda(G)-\frac{2e(G)}{3\lambda(G)}$ whenever $\lambda(G)^2>e(G)$. As the first consequence, we derive that an $n$-vertex graph $G$ with $e(G)>n^2/4$ satisfies that $\mathrm{bk} (G)>\frac{2e(G)}{n} - \frac{n}{3}$, which is stronger than an old conjecture of Erd\H{o}s, and was proved by Edwards. The second corollary is a positive solution to an open problem by Zhai and Lin (JGT, 2023). The third application is a positive solution to an open problem of Li, Liu, and Zhang (JCTB, 2026), which was independently proved by Zhang et al.
The $\text{bk}(G)$ of a graph $G$ is the maximum number of triangles sharing a common edge. Motivated by a classical conjecture of Erd\H{o}s, spectral lower bounds for the booksize have received considerable attention. For a positive divisor $s$ of $m-1$ with $\frac{m-1}{s}\ge2$, let $S_{m,s}^{+}$ be obtained from $K_{...
We prove the coarse Erd\H{o}s-P\'{o}sa conjecture of Georgakopoulos and Papasoglu. Informally, any graph either contains many fat cycles that are pairwise far apart, or there is a small number of bounded radius balls that together hit all of them. To be more precise, if $G$ is a graph with no $q$-fat model of $k \cdot...
Sandra Albrechtsen, Marthe Bonamy, Romain Bourneuf et al.· 0 citations
It is well known that the \Lovasz-Theta function of a random graph $G(n,\tfrac{1}{2})$ is $\Theta(\sqrt{n})$. More precisely, it is tightly concentrated in the interval \( [\sqrt{n},\, 2\sqrt{n}], \) where the upper bound follows from an explicit dual witness for the associated semidefinite program. Numerical evidence...
Let $G$ be a simple graph on $n$ vertices and $e(G)$ edges. Let $\mu_1\geq \cdots \geq \mu_{n-1}\geq \mu_n=0$ be the Laplacian eigenvalues of $G$. For $k=1, \ldots, n$, let $S_k(G)=\sum_{i=1}^{k}\mu_i$. Brouwers conjecture asserts that for any $k\in\{1,\ldots, n\}$, $S_k(G)\leq e(G)+\binom{k+1}{2}$. In [Bounding the su...
For graphs $H$ and $F$, let $ex(n,H,F)$ be the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. We study this problem when $H$ is a clique and $F=T_t$which is a fixed tree on $t$ vertices. The Erd\H{o}s--S\'{o}s conjecture concerns the value of $ex(n,K_2, T_t)$. Gerbner and Palmer proposed a more genera...
Given graphs $H$ and $F$, the generalized Tur\'{a}n number ${\rm ex}(n,H,F)$ is the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. Alon and Shikhelman (J. Combin. Theory Ser. B, 2016) initiated the systematic study of generalized Tur\'{a}n problems. Recently, Gao, Wu and Xue (J. Graph Theory, 2026) as...
Junpeng Zhou, Xi-Ying Yuan· 0 citations
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