Related papers: Trace Diagrams, Matrix Minors, and Determinant Ide…

This article provides structural characterization of simple graphs whose edge-set can be partitioned into maximum matchings. We use Vizing's classification of simple graphs based on edge chromatic index.

It is known that isomorphisms of graph Jacobians induce cyclic bijections on the associated graphs. We characterize when such cyclic bijections can be strengthened to graph isomorphisms, in terms of an easily computed divisor. The result…

We give an expression for the determinant of the twisted Laplacian associated with any linear representation of a finite quiver in terms of traces of the holonomy of its cycles. To establish this expression, we prove a general identity for…

In many ways, graphs are the main modality of data we receive from nature. This is due to the fact that most of the patterns we see, both in natural and artificial systems, are elegantly representable using the language of graph structures.…

A signed graph is a graph together with an assignment of signs to the edges. A closed walk in a signed graph is said to be positive (negative) if it has an even (odd) number of negative edges, counting repetition. Recognizing the signs of…

A connected graph is called a multi-block graph if each of its blocks is a complete multi-partite graph. Building on the work of \cite{Bp3,Hou3}, we compute the determinant and inverse of the distance matrix for a class of multi-block…

A signed graph is a graph whose edges are signed. In a vertex-signed graph the vertices are signed. The latter is called consistent if the product of signs in every circle is positive. The line graph of a signed graph is naturally…

We present sweeping line graphs, a generalization of $\Theta$-graphs. We show that these graphs are spanners of the complete graph, as well as of the visibility graph when line segment constraints or polygonal obstacles are considered. Our…

Graph neural networks are deep neural networks designed for graphs with attributes attached to nodes or edges. The number of research papers in the literature concerning these models is growing rapidly due to their impressive performance on…

A labelled, undirected graph is a graph whose edges have assigned labels, from a specific set. Given a labelled, undirected graph, the well-known minimum labelling spanning tree problem is aimed at finding the spanning tree of the graph…

It has been conjectured that if a finite graph has a vertex coloring such that the union of any two color classes induces a connected graph, then for every set $T$ of vertices containing exactly one member from each color class there exists…

Several matroids can be defined on the edge set of a graph. Although historically the cycle matroid has been the most studied, in recent times, the bicircular matroid has cropped up in several places. A theorem of Matthews from late 1970s…

We consider a modified notion of planarity, in which two nations of a map are considered adjacent when they share any point of their boundaries (not necessarily an edge, as planarity requires). Such adjacencies define a map graph. We give…

Formulas for matrix determinants, algebraic adjunctions, characteristic polynomial coefficients, components of eigenvectors are obtained in the form of signless sums of matrix elements products taking by special graphs. Signless formulas…

Directed acyclic graphs (DAGs) are a class of graphs commonly used in practice, with examples that include electronic circuits, Bayesian networks, and neural architectures. While many effective encoders exist for DAGs, it remains…

Recently, graphs have been widely used to represent many different kinds of real world data or observations such as social networks, protein-protein networks, road networks, and so on. In many cases, each node in a graph is associated with…

This note concerns a one-line diagrammatic proof of the Cayley-Hamilton Theorem. We discuss the proof's implications regarding the "core truth" of the theorem, and provide a generalization. We review the notation of trace diagrams and…

We characterise the structure of those graphs of a given order which maximise the number of connected induced subgraphs for seven different graph classes, each with other prescribed parameters like minimum degree, independence number,…

We describe varieties of minimal rational tangents on the wonderful symmetric varieties by marked Dynkin diagrams. An irreducible component of a variety of minimal rational tangents is a rational homogeneous space, and hence, we have a…

We organize a table of regular graphs with minimal diameters and minimal mean path lengths, large bisection widths and high degrees of symmetries, obtained by enumerations on supercomputers. These optimal graphs, many of which are newly…