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

Uniform Mixing and Association Schemes

We consider continuous-time quantum walks on distance-regular graphs of small diameter. Using results about the existence of complex Hadamard matrices in association schemes, we determine which of these graphs have quantum walks that admit uniform mixing. First we apply a result due to Chan to show that the only strongly regular graphs that admit instantaneous uniform mixing are the Paley graph of order nine and certain graphs corresponding to regular symmetric Hadamard matrices with constant diagonal. Next we prove that if uniform mixing occurs on a bipartite graph X with n vertices, then n is divisible by four. We also prove that if X is bipartite and regular, then n is the sum of two integer squares. Our work on bipartite graphs implies that uniform mixing does not occur on C_{2m} for m >= 3. Using a result of Haagerup, we show that uniform mixing does not occur on C_p for any prime p such that p >= 5. In contrast to this result, we see that epsilon-uniform mixing occurs on C_p for all primes p.Comment: 23 page

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

A quantum protocol to win the graph colouring game on all Hadamard graphs

This paper deals with graph colouring games, an example of pseudo-telepathy, in which two provers can convince a verifier that a graph $G$ is $c$-colourable where $c$ is less than the chromatic number of the graph. They win the game if they convince the verifier. It is known that the players cannot win if they share only classical information, but they can win in some cases by sharing entanglement. The smallest known graph where the players win in the quantum setting, but not in the classical setting, was found by Galliard, Tapp and Wolf and has 32,768 vertices. It is a connected component of the Hadamard graph $G_N$ with $N=c=16$. Their protocol applies only to Hadamard graphs where $N$ is a power of 2. We propose a protocol that applies to all Hadamard graphs. Combined with a result of Frankl, this shows that the players can win on any induced subgraph of $G_{12}$ having 1609 vertices, with $c=12$. Combined with a result of Frankl and Rodl, our result shows that all sufficiently large Hadamard graphs yield pseudo-telepathy games.Comment: 5pag

Tremain equiangular tight frames

Equiangular tight frames provide optimal packings of lines through the origin. We combine Steiner triple systems with Hadamard matrices to produce a new infinite family of equiangular tight frames. This in turn leads to new constructions of strongly regular graphs and distance-regular antipodal covers of the complete graph.Comment: 11 page