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Mathematics
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The Matrix Theory of Graphs
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The Matrix Theory of Graphs

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematics and Computer Science".

Deadline for manuscript submissions: closed (30 January 2023) | Viewed by 3017

Special Issue Editor

Prof. Dr. David Dolžan

E-Mail Website
Guest Editor
Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia
Interests: matrix theory; matrices; finite rings; semirings; graphs in connection with algebra

Special Issue Information

Dear Colleagues,

The topic of the matrix theory of graphs investigates the relationship between the graph theory and their associated matrix representations and it explores the matrix properties of the graphs from the point of view of linear algebra, as well as the consequences that these results have for the graphs themselves. This includes the study of

  • the adjacency matrices,
  • incidence matrices,
  • path matrices,
  • distance matrices, and
  • Laplacian matrices, etc.

On the other hand, one can also define a graph according to the properties of a certain algebraic structure (for example, the commuting graph or the zero-divisor graph). The most important examples of such graphs are prescribed to the full matrix ring (or semiring) and the group of invertible matrices over a ring (or semiring). The four main cases of research interest in this area are:

  • realizability,
  • the problem of uniqueness,
  • connectivity and diameter,
  • other properties (metric dimension, chromatic number, etc.).

Prof. Dr. David Dolžan
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • graph
  • matrix
  • commutativity

Published Papers (2 papers)

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Research

10 pages, 278 KiB  
Article
Linear Algebraic Relations among Cardinalities of Sets of Matroid Functions
by Martin Kochol
Mathematics 2023, 11(11), 2570; https://doi.org/10.3390/math11112570 - 3 Jun 2023
Cited by 1 | Viewed by 616
Abstract
We introduce a unifying approach for invariants of finite matroids that count mappings to a finite set. The aim of this paper is to show that if the cardinalities of mappings with fixed values on a restricted set satisfy contraction–deletion rules, then there [...] Read more.
We introduce a unifying approach for invariants of finite matroids that count mappings to a finite set. The aim of this paper is to show that if the cardinalities of mappings with fixed values on a restricted set satisfy contraction–deletion rules, then there is a relation among them that can be expressed in terms of linear algebra. In this way, we study regular chain groups, nowhere-zero flows and tensions on graphs, and acyclic and totally cyclic orientations of oriented matroids and graphs. Full article
(This article belongs to the Special Issue The Matrix Theory of Graphs)
11 pages, 357 KiB  
Article
On the Estrada Indices of Unicyclic Graphs with Fixed Diameters
by Wenjie Ning and Kun Wang
Mathematics 2021, 9(19), 2395; https://doi.org/10.3390/math9192395 - 26 Sep 2021
Viewed by 1310
Abstract
The Estrada index of a graph G is defined as EE(G)=∑i=1neλi, where λ1,λ2,…,λn are the eigenvalues of the adjacency matrix [...] Read more.
The Estrada index of a graph G is defined as EE(G)=∑i=1neλi, where λ1,λ2,…,λn are the eigenvalues of the adjacency matrix of G. A unicyclic graph is a connected graph with a unique cycle. Let U(n,d) be the set of all unicyclic graphs with n vertices and diameter d. In this paper, we give some transformations which can be used to compare the Estrada indices of two graphs. Using these transformations, we determine the graphs with the maximum Estrada indices among U(n,d). We characterize two candidate graphs with the maximum Estrada index if d is odd and three candidate graphs with the maximum Estrada index if d is even. Full article
(This article belongs to the Special Issue The Matrix Theory of Graphs)
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