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 examples arising from maximal real equiangular tight frames. In this paper, we characterize the equality case. We also obtain an explicit combinatorial description of the extremal graphs for $\lambda_3$ and $\lambda_4$.
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 $\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_n$ be the eigenvalues of a simple graph $G$ of order $n$. The HL-index of $G$ is defined by $R(G)=\max\|\lambda_h|,|\lambda_\ell|\}$ with $h=\lfloor(n+1)/2\rfloor$ and $\ell=\lceil(n+1)/2\rceil$.In this paper, we prove that if $G$ is $ K_4$-minor-free or $ K _ {2,3} $-mi...
For a simple graph $G$ of order $n$, let $S_2(G)=\lambda_1(G)+\lambda_2(G)$ denote its spectral sum. We determine, for every $n\geq5$, the exact maximum of $S_2(G)$ and all equality cases. The unique maximizer, up to isomorphism, is the complement of the disjoint union of a suitably balanced complete bipartite graph an...
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 $ \lambda_1(G)\ge \lambda_2(G)\ge \cdots \ge \lambda_n(G)$ denote the adjacency eigenvalues of $G$. We prove that for all looped graphs $G$ on $n$ vertices, \[\lambda_1(G) + \lambda_2(\overline{G}) \le \frac87 n. \] Our method also gives a short new proof of the Nordhaus-Gaddum result for the spect...
Sahil Agarwal, Carter Antley, Joseph Aulenbacher et al.· 1 citation· ⚡1
For a graph $G$ of order $n$, let $\mathcal E(G)$ denote its adjacency energy and let $\alpha(G)$ denote its independence number. A recent theorem of Kumar and Pragada states that $$\mathcal E(G)\ge 2\bigl(n-\alpha(G)\bigr).$$ We determine all graphs attaining equality. More precisely, equality holds if and only if eve...
S. A. Mojallal· 1 citation
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