We prove that every finite simple triangle-free graph $G$ on $n\geq1$ vertices satisfies $q_{\min}(G)\leq7n/50$, where $q_{\min}(G)$ is the least eigenvalue of its signless Laplacian matrix $Q_G=D_G+A_G$, with $D_G$ the diagonal matrix of vertex degrees and $A_G$ the adjacency matrix. The constant is sharp, since the H...
A graph is $K_{r+1}$-saturated if it is $K_{r+1}$-free and adding any missing edge creates a copy of $K_{r+1}$. Kim, Kim, Kostochka, and O conjectured that $K_{r-1}\vee(n-r+1)K_1$ minimizes the spectral radius among all $n$-vertex $K_{r+1}$-saturated graphs. They proved the case $r=2$, and the cases $r=3$ and $r\in\{4,...
For a finite simple graph $G$ and an integer $k\ge0$, let $w_k(G)$ denote the number of walks of length $k$. We prove the conjecture of T\"aubig, Weihmann, Kosub, Hemmecke, and Mayr for every finite tree and determine all equality cases. If $T$ has $n\ge1$ vertices, then $n w_{k+1}(T)-2(n-1)w_k(T)\ge0$ for every $k\ge1...
Dong-Xiu Cai, Zhen-Bo Chen, Jia-Sheng Zeng et al.· 0 citations
For a tree $T$, let $g(T)=\lambda_1(T)-\lambda_2(T)$ be the difference between its two largest adjacency eigenvalues. A balanced double comet is obtained by attaching equally many leaves to the two endpoints of a path. Jovovi\'c, Koledin and Stani\'c conjectured that such a tree attains the minimum adjacency spectral g...
The Grone--Merris inequality, conjectured by Grone and Merris~(1994) and first proved by Bai~(2011), states that for every graph $G$ of order $n$ and every $1\le k\le n$, $\sum_{i=1}^k\lambda_i(G)\le\sum_{i=1}^k d_i^*(G)$, where $\lambda_1\ge\cdots\ge\lambda_n$ are the Laplacian eigenvalues and $d_1^*\ge\cdots\ge d_n^*...
Dong-Xiu Cai, Zheng-Bo Chen, Jia Yang et al.· 1 citation
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