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 $G$, \[\mathcal{E}(G) \ge 2\left(n-\alpha(G)\right),\] where $\alpha(G)$ denotes the independence number. We prove this conjecture.
For $n>2$ and $k>1$, define the polynomial \[F_{n,k}(x) = \Phi_n(1 + x + \cdots + x^k),\] where $\Phi_n$ denotes the $n$-th cyclotomic polynomial. The \emph{cyclotomic conjecture} proposed by Gimbert (1999) exactly describes the irreducibility of $F_{n,k}(x)$ over $\mathbb{Q}$ in terms of $n$ and $k$. Conde, Gimbert, Gonz\'{a}lez, Miller and Miret (2014) established that the cyclotomic conjecture, if true, would imply the non-existence of almost Moore digraphs - a well-known open question concerning the directed degree-diameter problem. In this article, we prove the cyclotomic conjecture and, as a consequence, show that there are no almost Moore digraphs with maximum out-degree $d$ and diameter $k$ for any $d>1$ and $k>2$.
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 $\lambda_4$.
Hitesh Kumar, Bojan Mohar, S. A. Mojallal et al.· 0 citations
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