Related papers: On maximal distances in a commuting graph
The commuting variety of matrices over a given field is a well-studied object in linear algebra and algebraic geometry. As a set, it consists of all pairs of square matrices with entries in that field that commute with one another. In this…
We show that large subsets of vector spaces over finite fields determine certain point configurations with prescribed distance structure. More specifically, we consider the complete graph with vertices as the points of $A \subseteq…
We calculate the diameters of commuting graphs of matrices over the binary Boolean semiring, the tropical semiring and an arbitrary nonentire commutative semiring. We also find the lower bound for the diameter of the commuting graph of the…
For any sequence of matrix algebras that converge to a coadjoint orbit we give explicit formulas that show that the distances between the matrix algebras (viewed as quantum metric spaces) converges to 0. In the process we develop a general…
We study the maximal rank in affine subspaces of symmetric or alternating matrices, in terms of the matching numbers of certain associated graphs. Applications include simple proofs of upper bounds on the dimension of such subspaces in…
Let $G$ be a simple connected graph, and $D(G)$ be the distance matrix of $G$. Suppose that $D_{\max}(G)$ and $\lambda_1(G)$ are the maximum row sum and the spectral radius of $D(G)$, respectively. In this paper, we give a lower bound for…
The \emph{distance-number} of a graph $G$ is the minimum number of distinct edge-lengths over all straight-line drawings of $G$ in the plane. This definition generalises many well-known concepts in combinatorial geometry. We consider the…
We study the commuting graph of $n\times n$ matrices over the field of $p$-adics $\mathbb{Q}_p$, whose vertices are non-scalar $n\times n$ matrices with entries in $\mathbb{Q}_p$ and whose edges connect pairs of matrices that commute under…
We study a class of overdetermined algebraic systems of equations. We prove that the number of distinct solutions equals to the maximal possible if and only if certain matrices are commuting and semisimple. This gives a characterization of…
The commuting graph of a semigroup is the set of non-central elements; the edges are defined as pairs $(u,v)$ satisfying $uv=vu$. We provide an example of a field $F$ and an integer $n$ such that the commuting graph of…
Geometric graphs appear in many real-world data sets, such as road networks, sensor networks, and molecules. We investigate the notion of distance between embedded graphs and present a metric to measure the distance between two geometric…
A question going back to Halmos asks when two approximately commuting matrices of a certain kind are close to exactly commuting matrices of the same kind. It has long been known that there is a winding number obstruction for approximately…
We define a distance metric between partitions of a graph using machinery from optimal transport. Our metric is built from a linear assignment problem that matches partition components, with assignment cost proportional to transport…
For a connected graph $G$ and $\alpha\in [0,1)$, the distance $\alpha$-spectral radius of $G$ is the spectral radius of the matrix $D_{\alpha}(G)$ defined as $D_{\alpha}(G)=\alpha T(G)+(1-\alpha)D(G)$, where $T(G)$ is a diagonal matrix of…
Optimal transportation distances are valuable for comparing and analyzing probability distributions, but larger-scale computational techniques for the theoretically favorable quadratic case are limited to smooth domains or regularized…
Let $G$ be a connected graph with $V(G)=\{v_1,\ldots,v_n\}$. The $(i,j)$-entry of the distance matrix $D(G)$ of $G$ is the distance between $v_i$ and $v_j$. In this article, using the well-known Ramsey's theorem, we prove that for each…
We investigate properties which ensure that a given finite graph is the commuting graph of a group or semigroup. We show that all graphs on at least two vertices such that no vertex is adjacent to all other vertices is the commuting graph…
We establish maximal trees and graphs for the difference of average distance and proximity proving thus the corresponding conjecture posed in [4]. We also establish maximal trees for the difference of average eccentricity and remoteness and…
The paper is devoted to studying the orthogonality graph of the matrix ring over a commutative ring. It is proved that the orthogonality graph of the ring of matrices with size greater than 1 over a commutative ring with zero-divisors is…
We will give a short proof of the fact that if the algebraic closure of a field $\mathbb F$ is a finite extension, then for $n\geq 3$ the commuting graph $\Gamma(M_n(\mathbb F))$ is connected and its diameter is four.