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Register a new userMaximality of Seidel matrices and switching roots of graphs
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In this paper, we discuss maximality of Seidel matrices with a fixed largest eigenvalue. We present a classification of maximal Seidel matrices of largest eigenvalue $3$, which gives a classification of maximal equiangular lines in a Euclidean space with angle $arccos1/3$. Motivated by the maximality of the exceptional root system $E_8$, we define strong maximality of a Seidel matrix, and show that every Seidel matrix achieving the absolute bound is strongly maximal.
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In this short note we observe that recent results of Abert and Hubai and of Csikvari and Frenkel about Benjamini--Schramm continuity of the holomorphic moments of the roots of the chromatic polynomial extend to the theory of dense graph sequences. We offer a number of problems and conjectures motivated by this observation.
In 1960, Hoffman and Singleton cite{HS60} solved a celebrated equation for square matrices of order $n$, which can be written as $$ (kappa - 1) I_n + J_n - A A^{rm T} = A$$ where $I_n$, $J_n$, and $A$ are the identity matrix, the all one matrix, and a $(0,1)$--matrix with all row and column sums equal to $kappa$, respectively. If $A$ is an incidence matrix of some configuration $cal C$ of type $n_kappa$, then the left-hand side $Theta(A):= (kappa - 1)I_n + J_n - A A^{rm T}$ is an adjacency matrix of the non--collinearity graph $Gamma$ of $cal C$. In certain situations, $Theta(A)$ is also an incidence matrix of some $n_kappa$ configuration, namely the neighbourhood geometry of $Gamma$ introduced by Lef`evre-Percsy, Percsy, and Leemans cite{LPPL}. The matrix operator $Theta$ can be reiterated and we pose the problem of solving the generalised Hoffman--Singleton equation $Theta^m(A)=A$. In particular, we classify all $(0,1)$--matrices $M$ with all row and column sums equal to $kappa$, for $kappa = 3,4$, which are solutions of this equation. As a by--product, we obtain characterisations for incidence matrices of the configuration $10_3F$ in Kantors list cite{Kantor} and the $17_4$ configuration $#1971$ in Betten and Bettens list cite{BB99}.
For a given hypergraph, an orientation can be assigned to the vertex-edge incidences. This orientation is used to define the adjacency and Laplacian matrices. In addition to studying these matrices, several related structures are investigated including the incidence dual, the intersection graph (line graph), and the 2-section. A connection is then made between oriented hypergraphs and balanced incomplete block designs.
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.
We study the behaviour of the 2-rank of the adjacency matrix of a graph under Seidel and Godsil-McKay switching, and apply the result to graphs coming from graphical Hadamard matrices of order $4^m$. Starting with graphs from known Hadamard matrices of order $64$, we find (by computer) many Godsil-McKay switching sets that increase the 2-rank. Thus we find strongly regular graphs with parameters $(63,32,16,16)$, $(64,36,20,20)$, and $(64,28,12,12)$ for almost all feasible 2-ranks. In addition we work out the behaviour of the 2-rank for a graph product related to the Kronecker product for Hadamard matrices, which enables us to find many graphical Hadamard matrices of order $4^m$ for which the related strongly regular graphs have an unbounded number of different 2-ranks. The paper extends results from the article Switched symplectic graphs and their 2-ranks by the first and the last author.
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