Let $G$ be a graph of order $n$ with eigenvalues $\lambda_1(G) \geq \dots \geq \lambda_n(G)$, and let $s_+(G)=\sum_{\lambda_i(G)>0}\lambda_i(G)^2.$ Recently Liu, Tang, and Zhang proved the positive square-energy strengthening of Tur\'an's theorem \[\sqrt{s_+(G)}\leq \left(1-\frac1r\right)n.\] where $r=\omega(G)$ is the clique number of $G$. We characterize the families of graphs for which the above inequality is sharp. Precisely, we prove that, for $r\geq 2$, equality holds if and only if $r\mid n$ and $G$ is the complete regular $r$-partite graph $K_{n/r,\ldots,n/r}$.
Let $\Sigma=(G,\sigma)$ be a connected signed graph of order $n$ and size $m$, and let $s^{+}(\Sigma)$ and $s^{-}(\Sigma)$ denote the sums of the squares of its positive and negative adjacency eigenvalues, respectively. The square-energy conjecture of Elphick, Farber, Goldberg, and Wocjan states that every connected gr...
For a simple graph $G$ of order $n$, let $\lambda_1(G)\ge \cdots \ge \lambda_n(G)$ denote its adjacency eigenvalues. Hong's problem asks for the optimal upper bound for $\lambda_k(G)$. A recent theorem of Sivashankar gives, for every $k\ge3$, \[ \lambda_k(G)\le \frac{(k-2)\sqrt{k+1}+2}{2k(k-1)}\,n-1, \] with sharp exam...
Hitesh Kumar, Bojan Mohar, S. A. Mojallal et al.· 0 citations
For an integer $k\ge2$, let $\lambda_k(G)$ denote the $k$th largest adjacency eigenvalue of a graph $G$. For every graph $G$ on $n$ vertices and every $2 \leq k \leq n$, we prove \[ \lambda_k(G) \le \frac{(k-2)\sqrt{k+1}+2}{2k(k-1)}\,n-1. \] Our bound is tight for $k\in\{2,3,4,8,24\}$. We obtain it by reducing the grap...
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...
For a graph $G$ of order $n$, with adjacency eigenvalues $\lambda_1(G) \geq \cdots \geq \lambda_n(G)$, the \emph{energy} of $G$ is defined to be \[\mathcal{E}(G)=\sum_{i=1}^{n} |\lambda_i(G)|.\] A well-known conjecture from the 1980s by Fajtlowicz states that for any graph $G$, \[\mathcal{E}(G) \ge 2\left(n-\alpha(G)\r...
For a graph $G$, let $s^+(G)$ and $s^-(G)$ denote the sums of the squares of its positive and negative adjacency eigenvalues. We determine all equality cases in the conjecture of Elphick, Farber, Goldberg, and Wocjan that every connected graph $G$ on $n$ vertices satisfies \[ \min \{s^+(G),s^-(G)\}\ge n-1. \] Namely, e...
Fu-Tao Hu, Ya-Yang Liu, Yi Wang· 1 citation
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