Preface to the Special Issue “Algebraic Structures and Graph Theory, 2nd Edition”

This is a continuation of the work initiated in [1], representing a reprint of the second edition of the Special Issue “Algebraic Structures and Graph Theory”, which was published in the MDPI journal Mathematics. Among the 36 submissions received for this Special Issue, the editors selected ten articles and one review paper that successfully passed the peer-review process, and were then published in the journal in the period from March 2023 to November 2024. They contain original research ideas that have made a significant advancements in the theory of algebraic structures and graph theory. In particular, the topics discussed in these 11 papers are related to graphs constructed from lattices, semigroups, or groups, to particular types of graphs (as edge-primitive, friendship, or equitable graphs), and to hypercompositional algebras (HX-groups and multi-rings).

Contribution 1 proposes a characterization of the crosscap two annihilating ideals graphs of lattices with at most four atoms. As a consequence, a large class of r-partite graphs that can be embedded in the Klein bottle has been introduced. Contribution 2 discusses the planarity of $S(m,e)$-graphs associated with some irreducible numerical semigroups with multiplicity m and embedding dimension e. In Contribution 3, the authors characterize the maximal connected subdigraphs of the Cayley digraph of a Clifford semigroup related to one of its subsets. This study helps to investigate on the independence numbers of the Cayley digraphs of Clifford semigroups. Based on the properties of non-abelian simple groups having at least one subgroup of order $pq$, where p and q are two distinct odd primes, the authors of Contribution 4 completely determined the edge-primitive graphs of order $pq$. In Contribution 5 we can find a model for calculating the upper bounds of the radio numbers of the so-called friendship graphs having k cycles, each of length m, with $3\le m\le 6$, and having one common vertex. The study conducted in Contribution 6 leads to a structure theorem for semiconic idempotent commutative residuated lattices. This theorem is the key element to prove that the variety of strongly semiconic idempotent commutative residuated lattices has the amalgamation property. A classification of the seven-valent symmetric graphs of order $8pq$, where p and q are distinct primes, is presented in Contribution 7. The main idea used by the authors of this paper is the reduction of the automorphism groups of the considered graphs to some non-commutative simple groups. Another interesting connection between graphs and groups arises in Contribution 8. In this paper, the authors study some topological indices and graph-theoretic properties (such as connectedness, diameter, girth, clique number, and radius) of equitable graphs of type I constructed from various groups. The last two original articles within this Special Issue deal with algebraic multistructures. In Contribution 9, the concept of Marshall’s quotient of a non-commutative multi-ring with involution is studied, leading to new examples of multialgebras with involution. Contribution 10 presents an algorithm for computing the HX-groups that have support equal to the dihedral group $D_n$. We conclude this Special Issue with Contribution 11, which is a review paper on Feynam diagrams, introduced in quantum electrodynamics and also used in biology and economy nowadays. The main analytical and algebraic properties of these diagrams are summarized, with examples related to wave propagation, information field theory, and medicine.

The Guest Editors extend their sincere appreciation to all of the authors for their valuable contributions to this Special Issue. We are also deeply grateful to the anonymous reviewers for their insightful and professional evaluation reports, which have significantly enhanced the quality of the submitted manuscripts. Furthermore, we acknowledge the excellent collaboration with the publisher, the constant assistance provided by the MDPI associate editors in bringing this project to the end, and the great support of the Managing Editor of this Special Issue, Ms. Ursula Tian.