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Mathematics
Special Issues
Algebraic Combinatorics and Spectral Graph Theory
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Algebraic Combinatorics and Spectral Graph Theory

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "A: Algebra and Logic".

Deadline for manuscript submissions: closed (20 April 2025) | Viewed by 494

Special Issue Editors

Prof. Dr. Irene Sciriha

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Guest Editor
Department of Mathematics, University of Malta, MSD 2080 Msida, Malta
Interests: spectral graph theory; quantum mechanics; networks; carbon molecules; molecular conductivity
Special Issues, Collections and Topics in MDPI journals
Dr. John Gauci

E-Mail Website
Guest Editor
Department of Mathematics, University of Malta, MSD 2080 Msida, Malta
Interests: spectral graph theory; connectivity in graphs; crossing numbers

Special Issue Information

Dear Colleagues,

Algebraic combinatorics lies at the crossroads of algebra and combinatorics, exploiting the interaction between these two vast areas of mathematics. By combining tools and methods from both areas, discrete structures are studied through the application of either algebraic methods in combinatorial contexts or of combinatorial techniques in algebraic settings. A vast array of areas, including but not limited to representation theory, knot theory, symmetric functions, and mathematical physics, are closely linked with algebraic combinatorics. The combinatorial properties of discrete structures, in general, and of graphs, in particular, can be expressed using eigenvalues and eigenvectors of matrices related to these structures/graphs. This approach was first introduced in the late 1980s in an attempt to prove Cheeger’s inequality for finding a sparse cut. The most common matrices associated with a graph are the adjacency matrix and the Laplacian matrix. Once the eigenvalues and eigenvectors of such matrices are computed (or estimated), these can be related to structural properties of graphs. For example, it is a well-known result of Fiedler (1973) that the second smallest eigenvalue of the Laplacian matrix of a graph is positive if and only if the graph is connected. It is indeed the case that some of the most useful and beautiful applications of spectral graph theory are combinatorial.

Prof. Dr. Irene Sciriha
Dr. John Gauci
Guest Editors

Manuscript Submission Information

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Keywords

  • eigenvalues of a graph/digraph
  • spectral graph theory
  • sum of the k largest eigenvalues/singular values
  • energy of a graph/digraph
  • distance
  • distance energy
  • topological indices
  • extremal graphs/digraphs
  • spectral determination
  • complex networks
  • complex systems

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Published Papers (1 paper)

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Research

24 pages, 313 KiB  
Article
Common Neighborhood Energy of the Non-Commuting Graphs and Commuting Graphs Associated with Dihedral and Generalized Quaternion Groups
by Hanaa Alashwali and Anwar Saleh
Mathematics 2025, 13(11), 1834; https://doi.org/10.3390/math13111834 - 30 May 2025
Viewed by 168
Abstract
This paper explores the common neighborhood energy (ECN(Γ)) of graphs derived from the dihedral group D2n and generalized quaternion group Q4n, specifically the non-commuting graph (NCM-graph) and the commuting graph (CM-graph). [...] Read more.
This paper explores the common neighborhood energy (ECN(Γ)) of graphs derived from the dihedral group D2n and generalized quaternion group Q4n, specifically the non-commuting graph (NCM-graph) and the commuting graph (CM-graph). Studying graphs associated with groups offers a powerful approach to translating algebraic properties into combinatorial structures, enabling the application of graph-theoretic tools to understand group behavior. The common neighborhood energy, defined as the sum of the absolute values of the eigenvalues of the common neighborhood (CN) matrix, i.e., ∑i=1p|ζi|, where {ζi}i=1p are the CN eigenvalues, provides insights into the structural properties of these graphs. We derive explicit formulas for the CN characteristic polynomials and corresponding CN eigenvalues for both the NCM-graph and CM-graph as functions of n. Consequently, we establish closed-form expressions for the ECN of these graphs, which are parameterized by n. The validity of our theoretical results is confirmed through computational examples. This study contributes to the spectral analysis of algebraic graphs, demonstrating a direct connection between the group-theoretic structure of D2n and Q4n, as well as the combinatorial energy of their associated graphs, thus furthering the understanding of group properties through spectral graph theory. Full article
(This article belongs to the Special Issue Algebraic Combinatorics and Spectral Graph Theory)
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