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The strongly annihilating-ideal graph SAG (R) of a commutative unital ring R is a simple graph whose vertices are non-zero ideals of R with non-zero annihilator and there exists an edge between two distinct vertices if and only if each of them has a non-zero intersection with annihilator of the other one. In this paper, we compute twin-free clique number of SAG (R) and as an application strong metric dimension of SAG (R) is given. Moreover, we investigate the structures of strong resolving sets in SAG (R) to find forcing strong metric dimension in SAG (R) . [ABSTRACT FROM AUTHOR]

Consider a finite group G which acts on itself such that the relation ∼ on G, which is defined by x ∼ y iff x is a fixed point of y under this action, is both reflexive and symmetric. Let fix (G) represent the set of fixed points of G, i.e., fix (G) = { x ∈ G | x g = x for all g ∈ G } . We associate with G a new graph using G ∖ fix (G) as its set of vertices, connecting vertices x , y ∈ G ∖ fix (G) iff x is a fixed point of y. This graph is referred to as the stabilizing graph of G and is denoted by Δ s (G) . Additionally, the complement of the stabilizing graph Δ s (G) , denoted by ∇ s (G) , is named the nonstabilizing graph of G. The aim of this study is to analyze various properties of the nonstabilizing graph ∇ s (G) and to explore how this graph-theoretical concept relates to the broader understanding of group actions. [ABSTRACT FROM AUTHOR]

In this paper, we construct a family of quasi-strongly regular Cayley graphs which is defined on a finite group G with respect to a proper subgroup H of G and denoted by Γ H (G) . We also compute the full automorphism group and characterize various transitivity properties of Γ H (G) for any group G and for any subgroup H of G. [ABSTRACT FROM AUTHOR]

A subset C of the vertex set of a graph Γ is said to be (α , β) -regular if C induces an α -regular subgraph and every vertex outside C is adjacent to exactly β vertices in C. In particular, if C is an (α , β) -regular set in some Cayley sum graph of a finite group G with connection set S, then C is called an (α , β) -regular set of G. By Sq(G) and NSq(G) we mean the set of all square elements and non-square elements of G. As one of the main results in this note, we show that a subgroup H of a finite abelian group G is an (α , β) -regular set of G, for each 0 ≤ α ≤ | NSq (G) ∩ H | and 0 ≤ β ≤ L (H) , where L (H) = | H | , if Sq (G) ⊆ H and L (H) = | NSq (G) ∩ H | , otherwise. As a consequence we easily get that H is a (0, 1)-regular, if and only if either Sq (G) ⊆ H or NSq (G) ∩ H ≠ ∅ . The proof of this result is given by X. Ma, K. Wang, and Y. Yang in 2022, in a longer method. Also, X. Ma, M. Feng, and K. Wang in 2020, gave a sufficient and necessary condition for a subgroup H to be a (0, 1)-regular subgroup of an abelian group G. Our new result makes that result much easier to gain. Also, we consider the dihedral group G = D 2 n and for each subgroup H of G, by giving an appropriate connection set S, we determine each possibility for (α , β) , where H is an (α , β) -regular set of G. [ABSTRACT FROM AUTHOR]

For a commutative ring R with identity, the zero-divisor graph of R, denoted Γ (R) , is the simple graph whose vertices are the nonzero zero-divisors of R where two distinct vertices x and y are adjacent if and only if x y = 0 . In this paper, we are interested in partitioning the vertex set of Γ (R) into global defensive alliances for a finite commutative ring R. This problem has been well investigated in graph theory. Here we connected it with the ring theoretical context. We characterize various commutative finite rings for which the zero-divisor graph is partitionable into global defensive alliances. We also give several examples to illustrate the scopes and limits of our results. [ABSTRACT FROM AUTHOR]

A nut graph is a simple graph of order 2 or more for which the adjacency matrix has a single zero eigenvalue such that all nonzero kernel eigenvectors have no zero entry (i.e. are full). It is shown by construction that every finite group can be represented as the group of automorphisms of infinitely many nut graphs. It is further shown that such nut graphs exist even within the class of regular graphs; the cases where the degree is 8, 12, 16, 20 or 24 are realised explicitly. [ABSTRACT FROM AUTHOR]

AbstractWe introduce the normalizing graph of a group and study the connectivity of the normalizing and permuting graphs of a group when the group is finite and soluble. In particular, we classify finite soluble groups with disconnected normalizing graph. The main results shows that if a finite soluble group has connected normalizing graph then this graph has diameter at most 6. Furthermore, this bound is tight. A corollary then presents the connectivity properties of the permuting graph. [ABSTRACT FROM AUTHOR]

The fixing number of a graph Γ is the minimum number of vertices that, when fixed, remove all nontrivial automorphisms from the automorphism group of Γ . Gibbons and Laison extended the concept of fixing numbers of graphs to fixing sets of groups. The fixing set of a finite group G is the set of all fixing numbers of graphs whose automorphism groups are isomorphic to G. For n ∈ Z with n ≥ 2 , Gibbons and Laison conjectured that the fixing set of the symmetric group on n symbols is { 1 , 2 , … , n - 1 } . In this article, we will disprove this conjecture, which also answers an open question of theirs. Moreover, we will establish previously unknown elements of fixing sets of symmetric groups by using Johnson graphs. Fixing sets of other groups have also been considered; however, there are very few fixing sets of groups that have been completely established. We will begin this article by proving the fixing sets of both quasi-dihedral groups and quasi-abelian groups; these results then determine the fixing set of every group that is isomorphic to a member in one of the six infinite families of 2-groups that contain a cyclic subgroup of index 2. [ABSTRACT FROM AUTHOR]

A finite group G is an exact product of two subgroups A and B if G = AB and A ∩ B = { 1 G } . In this paper, for all odd numbers k ≥ 3 , using the theory of skew morphisms and associated extended power functions, we present a classification of exact products of a cyclic group and a dihedral group D 2 k . As an application the regular generalized Cayley maps of odd valency over cyclic groups are classified. [ABSTRACT FROM AUTHOR]

We describe the full automorphism group of the directed reduced power graph and the undirected reduced power graph of a finite group. We compute the full automorphism groups of these graphs of several classes of finite groups. Also, we establish some relation between the automorphism group of the undirected reduced power graph (resp. the directed reduced power graph) and the automorphism group of the undirected power graph (resp. the directed power graph) of a finite group. [ABSTRACT FROM AUTHOR]

This work classifies three-dimensional simple evolution algebras over arbitrary fields. For this purpose, we use tools such as the associated directed graph, the moduli set, inductive limit group, Zariski topology and the dimension of the diagonal subspace. Explicitly, in the three-dimensional case, we construct some models $ _i\mathbf {III}_{\lambda _1,\ldots,\lambda _n}^{p,q} $ i III λ 1 , ... , λ n p , q of such algebras with $ 1\le i\le 4 $ 1 ≤ i ≤ 4 , $ \lambda _i\in {\mathbb {K}}^\times $ λ i ∈ K × , $ p,q\in {\mathbb {N}} $ p , q ∈ N , such that any algebra is isomorphic to one (and only one) of the given in the models and we further investigate the isomorphic question within each one. Moreover, we show how to construct simple evolution algebras of higher-order from known simple evolution algebras of smaller size. [ABSTRACT FROM AUTHOR]

Let G be a simple graph on the vertex set { v 1 , … , v n } . An algebraic object attached to G is the toric ideal I G . We say that I G is subgraph splittable if there exist subgraphs G 1 and G 2 of G such that I G = I G 1 + I G 2 , where both I G 1 and I G 2 are not equal to I G . We show that I G is subgraph splittable if and only if it is edge splittable. We also prove that the toric ideal of a complete bipartite graph is not subgraph splittable. In contrast, we show that the toric ideal of a complete graph K n is always subgraph splittable when n ≥ 4 . Additionally, we show that the toric ideal of K n has a minimal splitting if and only if 4 ≤ n ≤ 5 . Finally, we prove that any minimal splitting of I G is also a reduced splitting. [ABSTRACT FROM AUTHOR]

A graph Γ is G-symmetric if it admits G as a group of automorphisms acting transitively on the set of arcs of Γ , where an arc is an ordered pair of adjacent vertices. Let Γ be a G-symmetric graph such that its vertex set admits a nontrivial G-invariant partition B , and let D (Γ , B) be the incidence structure with point set B and blocks { B } ∪ Γ B (α) , for B ∈ B and α ∈ B , where Γ B (α) is the set of blocks of B containing at least one neighbor of α in Γ . In this paper, we classify all G-symmetric graphs Γ such that Γ B (α) ≠ Γ B (β) for distinct α , β ∈ B , the quotient graph of Γ with respect to B is a complete graph, and D (Γ , B) is isomorphic to the complement of a (G, 2)-point-transitive linear space. [ABSTRACT FROM AUTHOR]

AbstractRecently, Maleki determined the dimension of the Terwilliger algebra of Cn⋊C2, where Cn⋊C2=〈a,b|an=b2=1,bab−1=as〉. Subsequently, the authors determined the dimension of the Terwilliger algebra of Tn,s, where Tn,s=〈a,b|a2n=1,an=b2,bab−1=as〉. In this paper, generalizing these two results, we consider the Terwilliger algebras of the generalized dihedral groups and non-abelian finite groups admitting a cyclic subgroup of index 2. Moreover, we give a complete characterization of the Wedderburn components of the Terwilliger algebras of these groups. [ABSTRACT FROM AUTHOR]

A graph Γ is said to be 2-arc-transitive if its automorphism group acts transitively on the set of 2-arcs of Γ . In this paper, we give a group-theoretic characterization of those connected 2-arc-transitive graphs which admit a vertex-transitive simple group. [ABSTRACT FROM AUTHOR]

In the combinatorial context, one of the key problems in sequence reconstruction is to find the largest intersection of two metric balls of radius r. In this paper we study this problem for permutations of length n distorted by Hamming errors and determine the size of the largest intersection of two metric balls with radius r whose centers are at distance d = 2 , 3 , 4 . Moreover, it is shown that for any n ⩾ 3 an arbitrary permutation is uniquely reconstructible from four distinct permutations at Hamming distance at most two from the given one, and it is proved that for any n ⩾ 4 an arbitrary permutation is uniquely reconstructible from 4 n - 5 distinct permutations at Hamming distance at most three from the permutation. It is also proved that for any n ⩾ 5 an arbitrary permutation is uniquely reconstructible from 7 n 2 - 31 n + 37 distinct permutations at Hamming distance at most four from the permutation. Finally, in the case of at most r Hamming errors and sufficiently large n, it is shown that at least Θ (n r - 2) distinct erroneous patterns are required in order to reconstruct an arbitrary permutation. [ABSTRACT FROM AUTHOR]

Given a finite group GG with identity ee, the TI-power graph (trivial intersection power graph) defined on GG, denoted by Γ(G)\Gamma \left(G), is an undirected graph with vertex set GG where distinct vertices aa and bb are adjacent if ⟨a⟩∩⟨b⟩={e}\langle a\rangle \cap \langle b\rangle =\left\{e\right\}. We classify all finite groups whose TI-power graph is claw-free, K1,4{K}_{1,4}-free, C4{C}_{4}-free, and P4{P}_{4}-free. In addition, we classify the finite groups whose TI-power graph is a threshold graph, a cograph, a chordal graph, and a split graph.

The aim of this paper is to study the power graph of a semigroup. We obtain sufficient and necessary conditions for the independence number of the power graph of an (IC) abundant semigroups (superabundant semigroups) to be finite.