More eigenvalue problems of Nordhaus-Gaddum type

Let $G$ be a graph of order $n$ and let $\mu_{1}\left(G\right) \geq \cdots\geq\mu_{n}\left(G\right)$ be the eigenvalues of its adjacency matrix. This note studies eigenvalue problems of Nordhaus-Gaddum type. Let $\overline{G}$ be the complement of a graph $G.$ It is shown that if $s\geq2$ and $n\geq15\left(s-1\right) ,$ then $$\left\vert \mu_{s}\left(G\right) \right\vert +|\mu_{s}(\overline{G})|\,\leq n/\sqrt{2\left(s-1\right)}-1.$$ Also if $s\geq1$ and $n\geq4^{s},$ then $$\left\vert \mu_{n-s+1}\left(G\right) \right\vert +|\mu_{n-s+1}(\overline {G})|\,\leq n/\sqrt{2s}+1.$$ If $s=2^{k}+1$ for some integer $k$, these bounds are asymptotically tight. These results settle infinitely many cases of a general open problem.Comment: 12 pages, 0 figure

Extrema of graph eigenvalues

In 1993 Hong asked what are the best bounds on the $k$'th largest eigenvalue $\lambda_{k}(G)$ of a graph $G$ of order $n$. This challenging question has never been tackled for any $2<k<n$. In the present paper tight bounds are obtained for all $k>2,$ and even tighter bounds are obtained for the $k$'th largest singular value $\lambda_{k}^{\ast}(G).$ Some of these bounds are based on Taylor's strongly regular graphs, and other on a method of Kharaghani for constructing Hadamard matrices. The same kind of constructions are applied to other open problems, like Nordhaus-Gaddum problems of the kind: How large can $\lambda_{k}(G)+\lambda_{k}(\bar{G})$ be$?$ These constructions are successful also in another open question: How large can the Ky Fan norm $\lambda_{1}^{\ast}(G)+...+\lambda_{k}^{\ast }(G)$ be $?$ Ky Fan norms of graphs generalize the concept of graph energy, so this question generalizes the problem for maximum energy graphs. In the final section, several results and problems are restated for $(-1,1)$-matrices, which seem to provide a more natural ground for such research than graphs. Many of the results in the paper are paired with open questions and problems for further study.Comment: 32 page

Beyond graph energy: norms of graphs and matrices

In 1978 Gutman introduced the energy of a graph as the sum of the absolute values of graph eigenvalues, and ever since then graph energy has been intensively studied. Since graph energy is the trace norm of the adjacency matrix, matrix norms provide a natural background for its study. Thus, this paper surveys research on matrix norms that aims to expand and advance the study of graph energy. The focus is exclusively on the Ky Fan and the Schatten norms, both generalizing and enriching the trace norm. As it turns out, the study of extremal properties of these norms leads to numerous analytic problems with deep roots in combinatorics. The survey brings to the fore the exceptional role of Hadamard matrices, conference matrices, and conference graphs in matrix norms. In addition, a vast new matrix class is studied, a relaxation of symmetric Hadamard matrices. The survey presents solutions to just a fraction of a larger body of similar problems bonding analysis to combinatorics. Thus, open problems and questions are raised to outline topics for further investigation.Comment: 54 pages. V2 fixes many typos, and gives some new materia

Laplacian Distribution and Domination

Let $m_G(I)$ denote the number of Laplacian eigenvalues of a graph $G$ in an interval $I$, and let $\gamma(G)$ denote its domination number. We extend the recent result $m_G[0,1) \leq \gamma(G)$, and show that isolate-free graphs also satisfy $\gamma(G) \leq m_G[2,n]$. In pursuit of better understanding Laplacian eigenvalue distribution, we find applications for these inequalities. We relate these spectral parameters with the approximability of $\gamma(G)$, showing that $\frac{\gamma(G)}{m_G[0,1)} \not\in O(\log n)$. However, $\gamma(G) \leq m_G[2, n] \leq (c + 1) \gamma(G)$ for $c$-cyclic graphs, $c \geq 1$. For trees $T$, $\gamma(T) \leq m_T[2, n] \leq 2 \gamma(G)$