A conference matrix of order $n$ is an $n\times n$ matrix $C$ with diagonal entries $0$ and off-diagonal entries $\pm 1$ satisfying $CC^\top=(n-1)I$. If $C$ is symmetric, then $C$ has a symmetric spectrum $Σ$ (that is, $Σ=-Σ$) and eigenvalues $\pm\sqrt{n-1}$. We show that many principal submatrices of $C$ also have symmetric spectrum, which leads to examples of Seidel matrices of graphs (or, equivalently, adjacency matrices of complete signed graphs) with a symmetric spectrum. In addition, we show that some Seidel matrices with symmetric spectrum can be characterized by this construction.
We construct new, previously unknown parametric families of complex conference matrices and of complex Hadamard matrices of square orders and related them to complex equiangular tight frames.
A two-intersection set with parameters $(j;α,β)$ for a block design is a $j$-subset of the point set of the design, which intersects every block in $α$ or $β$ points. In this paper, we show the existence of a two-intersection set with parameters $(2m^2-m+1;m^2-m,m^2)$ for the block design obtained from translations of the set of nonzero squares in the finite field of order $q=4m^2+1$. As an application, we give a construction of conference matrices with maximum excess based on the two-intersection sets.
We study the existence and construction of circulant matrices $C$ of order $n\geq2$ with diagonal entries $d\geq0$, off-diagonal entries $\pm1$ and mutually orthogonal rows. These matrices generalize circulant conference ($d=0$) and circulant Hadamard ($d=1$) matrices. We demonstrate that matrices $C$ exist for every order $n$ and for $d$ chosen such that $n=2d+2$, and we find all solutions $C$ with this property. Furthermore, we prove that if $C$ is symmetric, or $n-1$ is prime, or $d$ is not an odd integer, then necessarily $n=2d+2$. Finally, we conjecture that the relation $n=2d+2$ holds fo…
We show that the existence of $\{\pm 1\}$-matrices having largest possible determinant is equivalent to the existence of certain tournament matrices. In particular, we prove a recent conjecture of Armario. We also show that large submatrices of conference matrices are determined by their spectrum.
In 1992, Godsil and Hensel published a ground-breaking study of distance-regular antipodal covers of the complete graph that, among other things, introduced an important connection with equi-isoclinic subspaces. This connection seems to have been overlooked, as many of its immediate consequences have never been detailed in the literature. To correct this situation, we first describe how Godsil and Hensel's machine uses representation theory to construct equi-isoclinic tight fusion frames. Applying this machine to Mathon's construction produces $q+1$ planes in $\mathbb{R}^{q+1}$ for any even pr…
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 pr…
We consider bordered complex Hadamard matrices whose core is contained in the Bose-Mesner algebra of a strongly regular graph. Examples include a complex Hadamard matrix whose core is contained in the Bose-Mesner algebra of a conference graph due to J. Wallis, F. Szöllősi, and a family of Hadamard matrices given by Singh and Dubey. In this paper, we prove that there are no other bordered complex Hadamard matrices whose core is contained in the Bose-Mesner algebra of a strongly regular graph.
We have extended the Paley constructions for Hadamard matrices and obtained some series of Hadamard matrices. Especially Paley construction-II is applicable for odd prime power q is congruent to 1(mod 4) however our method is applicable for any odd prime power. Some of which are non-isomorphic to the matrices obtained by standard Paley constructions, Sylvester construction, Williamson construction or combination of either of these constructions with Sylvester construction. In fact we have used the conference matrix of order prime power q in some different manner which is applicable for any odd…
We prove that symmetry group of the pfaffian polynomial of a symmetric matrix is a dihedral group. We calculate pfaffians of symmetric matrices with components $(x_i-x_j)^2$ and $\cos(x_i-x_j)$ for $i<j.$
We show that every principal submatrix of a square matrix $A$ with only entries in $\{0,1,-1\}$ has even rank if and only if $A$ is skew-symmetric up to multiplication of rows/columns by $-1$.
交替符号矩阵是满足以下条件的方阵：（i）所有元素都等于1、-1或0;（ii）每行和每列的和为1;（iii）每行和每列的非零元素符号交替。正方形的8元对称群以一种明显的方式作用于方阵。对于正方形的对称群的任何子群，我们可以考虑在这个子群的元素下不变的矩阵子集。这些子群有8个共轭类，从而产生8个矩阵对称类。R. P. Stanley建议研究这些对称类中的交替符号矩阵。我们已经发现的证据表明，六个对称类存在简单的产品公式的数量交替符号矩阵的类。此外，某些生成函数的因式分解指出了几个对称类与循环对称平面分区之间的惊人联系。
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: …
It is proved that a code $L(q)$ which is monomially equivalent to the Pless symmetry code $C(q)$ of length $2q+2$ contains the (0,1)-incidence matrix of a Hadamard 3-$(2q+2,q+1,(q-1)/2)$ design $D(q)$ associated with a Paley-Hadamard matrix of type II. Similarly, any ternary extended quadratic residue code contains the incidence matrix of a Hadamard 3-design associated with a Paley-Hadamard matrix of type I. If $q=5, 11, 17, 23$, then the full permutation automorphism group of $L(q)$ coincides with the full automorphism group of $D(q)$, and a similar result holds for the ternary extended quadr…
Finite quasi semimetrics on $n$ can be thought of as nonnegative valuations on the edges of a complete directed graph on $n$ vertices satisfying all possible triangle inequalities. They comprise a polyhedral cone whose symmetry groups were studied for small $n$ by Deza, Dutour and Panteleeva. We show that the symmetry and combinatorial symmetry groups are as they conjectured. Integral quasi semimetrics have apecial place in the theory of tiled orders, being known as exponent matrices, and can be viewed as monoids under componentwise maximum; we provide a novel derivation of the automorphism gr…
The \textit{eccentricity matrix} $\mathcal{E}(G)$ of a connected graph $G$ is obtained from the distance matrix of $G$ by keeping the largest non-zero entries in each row and each column, and leaving zeros in the remaining ones. The eigenvalues of $\mathcal{E}(G)$ are the \textit{$\mathcal{E}$-eigenvalues} of $G$. In this article, we find the inertia of the eccentricity matrices of trees. Interestingly, any tree on more than $4$ vertices with odd diameter has two positive and two negative $\mathcal{E}$-eigenvalues (irrespective of the structure of the tree). A tree with even diameter has the s…
We prove refined enumeration results on several symmetry classes as well as related classes of alternating sign matrices with respect to classical boundary statistics, using the six-vertex model of statistical physics. More precisely, we study vertically symmetric, vertically and horizontally symmetric, vertically and horizontally perverse, off-diagonally and off-antidiagonally symmetric, vertically and off-diagonally symmetric, quarter turn symmetric as well as quasi quarter turn symmetric alternating sign matrices. Our results prove conjectures of Fischer, Duchon and Robbins.
The nineteen-vertex model on a periodic lattice with an anti-diagonal twist is investigated. Its inhomogeneous transfer matrix is shown to have a simple eigenvalue, with the corresponding eigenstate displaying intriguing combinatorial features. Similar results were previously found for the same model with a diagonal twist. The eigenstate for the anti-diagonal twist is explicitly constructed using the quantum separation of variables technique. A number of sum rules and special components are computed and expressed in terms of Kuperberg's determinants for partition functions of the inhomogeneous…
