Max k-cut and the smallest eigenvalue

Let $G$ be a graph of order $n$ and size $m$, and let $\mathrm{mc}_{k}\left( G\right)$ be the maximum size of a $k$-cut of $G.$ It is shown that $$\mathrm{mc}_{k}\left( G\right) \leq\frac{k-1}{k}\left( m-\frac{\mu_{\min }\left( G\right) n}{2}\right) ,$$ where $\mu_{\min}\left( G\right)$ is the smallest eigenvalue of the adjacency matrix of $G.$ An infinite class of graphs forcing equality in this bound is constructed.Comment: 5 pages. Some typos corrected in v

The trace norm of r-partite graphs and matrices

The trace norm $\left\Vert G\right\Vert _{\ast}$ of a graph $G$ is the sum of its singular values, i.e., the absolute values of its eigenvalues. The norm $\left\Vert G\right\Vert _{\ast}$ has been intensively studied under the name of graph energy, a concept introduced by Gutman in 1978. This note studies the maximum trace norm of $r$-partite graphs, which raises some unusual problems for $r>2$. It is shown that, if $G$ is an $r$-partite graph of order $n,$ then $$\left\Vert G\right\Vert _{\ast}<\frac{n^{3/2}}{2}\sqrt{1-1/r}+\left( 1-1/r\right) n.$$ For some special $r$ this bound is tight: e.g., if $r$ is the order of a symmetric conference matrix, then, for infinitely many $n,$ there is a graph $G\$of order $n$ with $$\left\Vert G\right\Vert _{\ast}>\frac{n^{3/2}}{2}\sqrt{1-1/r}-\left( 1-1/r\right) n.$$Comment: 12 page

Remarks on the energy of regular graphs

The energy of a graph is the sum of the absolute values of the eigenvalues of its adjacency matrix. This note is about the energy of regular graphs. It is shown that graphs that are close to regular can be made regular with a negligible change of the energy. Also a $k$-regular graph can be extended to a $k$-regular graph of a slightly larger order with almost the same energy. As an application, it is shown that for every sufficiently large $n,$ there exists a regular graph $G$ of order $n$ whose energy $\left\Vert G\right\Vert_{\ast}$ satisfies $$\left\Vert G\right\Vert_{\ast}>\frac{1}{2}n^{3/2}-n^{13/10}.$$ Several infinite families of graphs with maximal or submaximal energy are given, and the energy of almost all regular graphs is determined.Comment: 12 pages. V2 corrects a typo. V3 corrects Theorem 1

Merging the A- and Q-spectral theories

Let $G$ be a graph with adjacency matrix $A\left( G\right)$, and let $D\left( G\right)$ be the diagonal matrix of the degrees of $G.$ The signless Laplacian $Q\left( G\right)$ of $G$ is defined as $Q\left( G\right) :=A\left( G\right) +D\left( G\right)$. Cvetkovi\'{c} called the study of the adjacency matrix the $A$% \textit{-spectral theory}, and the study of the signless Laplacian--the $Q$\textit{-spectral theory}. During the years many similarities and differences between these two theories have been established. To track the gradual change of $A\left( G\right)$ into $Q\left( G\right)$ in this paper it is suggested to study the convex linear combinations $A_{\alpha }\left( G\right)$ of $A\left( G\right)$ and $D\left( G\right)$ defined by $$A_{\alpha}\left( G\right) :=\alpha D\left( G\right) +\left( 1-\alpha\right) A\left( G\right) \text{, \ \ }0\leq\alpha\leq1.$$ This study sheds new light on $A\left( G\right)$ and $Q\left( G\right)$, and yields some surprises, in particular, a novel spectral Tur\'{a}n theorem. A number of challenging open problems are discussed.Comment: 26 page

Hypergraphs and hypermatrices with symmetric spectrum

It is well known that a graph is bipartite if and only if the spectrum of its adjacency matrix is symmetric. In the present paper, this assertion is dissected into three separate matrix results of wider scope, which are extended also to hypermatrices. To this end the concept of bipartiteness is generalized by a new monotone property of cubical hypermatrices, called odd-colorable matrices. It is shown that a nonnegative symmetric $r$-matrix $A$ has a symmetric spectrum if and only if $r$ is even and $A$ is odd-colorable. This result also solves a problem of Pearson and Zhang about hypergraphs with symmetric spectrum and disproves a conjecture of Zhou, Sun, Wang, and Bu. Separately, similar results are obtained for the $H$-spectram of hypermatrices.Comment: 17 pages. Corrected proof on p. 1

An asymptotically tight bound on the Q-index of graphs with forbidden cycles

Let G be a graph of order n and let q(G) be that largest eigenvalue of the signless Laplacian of G. In this note it is shown that if k>1 and q(G)>=n+2k-2, then G contains cycles of length l whenever 2<l<2k+3. This bound is asymptotically tight. It implies an asymptotic solution to a recent conjecture about the maximum q(G) of a graph G with no cycle of a specified length.Comment: 10 pages. Version 2 takes care of some mistakes in version