Search

For a simple graph G with edge ideal I(G), we study the N -graded Betti numbers in the linear strand of the minimal free resolution of I (Γ (Z n)) , where Γ (Z n) is the zero divisor graph of the ring Z n . We present sharp bounds for the Betti numbers of Γ (Z n) and characterize the graphs attaining these bounds, thereby establishing the correct equality case for one of the results of the earlier published paper (Theorem 4.5, S. Pirzada and S. Ahmad, On the linear strand of edge ideals of some zero divisor graphs, Commun. Algebra 51(2) (2023) 620–632). Also, we present homological invariants of the edge rings of Γ (Z n) for n = p 2 q and pqr, with primes p < q < r. [ABSTRACT FROM AUTHOR]

For a simple graph G with edge ideal I(G), we study the N -graded Betti numbers in the linear strand of the minimal free resolution of I (Γ (Z n)) , where Γ (Z n) is the zero divisor graph of the ring Z n . We present sharp bounds for the Betti numbers of Γ (Z n) and characterize the graphs attaining these bounds, thereby establishing the correct equality case for one of the results of the earlier published paper (Theorem 4.5, S. Pirzada and S. Ahmad, On the linear strand of edge ideals of some zero divisor graphs, Commun. Algebra 51(2) (2023) 620–632). Also, we present homological invariants of the edge rings of Γ (Z n) for n = p 2 q and pqr, with primes p < q < r. [ABSTRACT FROM AUTHOR]

AbstractLet S⊆N be a numerical semigroup with multiplicity m=min(S∖{0}) and conductor c=max(Z∖S)+1. Let P be the set of primitive elements, i.e. minimal generators, of S, and let L be the set of elements of S which are smaller than c. Wilf’s conjecture (1978) states that the inequality |P||L|≥c must hold. The conjecture has been shown to hold in case |P|≥m/2 by Sammartano in 2012, and subsequently in case |P|≥m/3 by the author in 2020. The main result in this paper is that Wilf’s conjecture holds in case |P|≥m/4 when m divides c. The proof uses divsets X, i.e. finite divisor-closed sets of monomials, as abstract models of numerical semigroups, and proceeds with estimates of the vertex-maximal matching number of the associated graph G(X). [ABSTRACT FROM AUTHOR]

AbstractThe inclusion graph of a group G, written as In(G), is defined to be an undirected graph with all nontrivial subgroups of G as its vertices, and two distinct subgroups H, K of G are adjacent if either H≤K or K≤H. A local inclusion graph of G with respect to a given subgroup S, written as In(G,S), is an induced subgraph of In(G) with vertex set consisting of all nontrivial subgroups of G properly containing S. Indeed, In(G) can be viewed as the special case when S is the identity subgroup. Let GLn(F) be the general linear group consisting of all n×n invertible matrices over a field F and Tn(F) the subgroup of all invertible lower triangular matrices in GLn(F). In this article, an arbitrary automorphism of In(GLn(F),Tn(F)) is determined and it is proved that the automorphism group of In(GLn(F),Tn(F)) is isomorphic to the direct product of Sn−1 and S2, where Sn−1 is the symmetric group of degree n−1. [ABSTRACT FROM AUTHOR]

AbstractLet G be a finite group and let e be its identity element. The intersection power graph of G is the undirected graph with vertex set G, in which two distinct vertices x and y are adjacent if either 〈x〉∩〈y〉≠{e} or one of x and y is e. In this paper, we first compute the strong metric dimension of the intersection power graph of a cyclic group, a dihedral group, and a generalized quaternion group. We also characterize all finite groups G whose intersection power graph has strong metric dimension |G|−2. Then, we give the sharp upper and lower bounds for the metric dimension of the intersection power graph of a finite group. As applications, we obtain the metric dimension of the intersection power graph of a cyclic group, a dihedral group, a generalized quaternion group, and a group of odd order. [ABSTRACT FROM AUTHOR]

Let S be a groupoid (magma) with zero 0, and let R = ⊕ s ∈ S R s be a contracted S-graded ring, that is, an S-graded ring with R 0 = 0. By G (H R) we denote the undirected power graph of a multiplicative subsemigroup H R = ∪ s ∈ S R s of R, and by G * (H R) a graph obtained from G (H R) by removing 0 and its incident edges. If Re is a nonzero ring component of R, then G * (R e) denotes a subgraph of G * (H R) , induced by R e *. In this paper we address a problem raised in [Abawajy, J., Kelarev, A., Chowdhury, M.: Power Graphs: A Survey. Electron. J. Graph Theory Appl. 1(2), 125–147 (2013)]. Namely, let S be torsion-free, that is, s n = t n implies s = t for all s, t ∈ S , and all positive integers n, and let S be 0-cancellative, that is, for all s, t, u ∈ S , su = tu ≠ 0 implies s = t , and us = ut ≠ 0 implies s = t. Also, let R be semisimple Artinian. We prove that if G * (R e) is connected for every nonzero ring component Re of R, then the connected components of G * (H R) are precisely the graphs G * (R e). [ABSTRACT FROM AUTHOR]

We construct a locally 5-arc transitive G-graph Δ, where G ∈ { PSU 3 (8) ⋊ C 2 , PSU 3 (8) ⋊ C 6 } . The corresponding vertex stabilizer amalgams are of local characteristic 3 but are not weak (B, N)-pairs. They are the first examples of this kind where there exists a vertex z such that the extension of G z / O p (G z [ 1 ]) over O p (G z [ 1 ]) is non-split. [ABSTRACT FROM AUTHOR]

AbstractConsider 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 | xg=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 Δs(G). Additionally, the complement of the stabilizing graph Δs(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]

AbstractIn 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 ΓH(G) . We also compute the full automorphism group and characterize various transitivity properties of ΓH(G) for any group G and for any subgroup H of G. [ABSTRACT FROM AUTHOR]

Let G be a finite non-abelian simple group, C a non-identity conjugacy class of G, and ΓC the Cayley graph of G based on C ∪ C − 1 . Our main result shows that in any such graph, there is an involution at bounded distance from the identity. [ABSTRACT FROM AUTHOR]

AbstractA finite group G is an exact product of two subgroups A and B if G = AB and A∩B={1G}. 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 D2k. As an application the regular generalized Cayley maps of odd valency over cyclic groups are classified. [ABSTRACT FROM AUTHOR]

AbstractWe 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]

Given a finite group G, denote by cs(G) the set of the sizes of the conjugacy classes of G and by Cent(G) the set of the centralizers of elements of G. Consider a prime p and integers s ≥ 2 and n ≥ 2 , with gcd (p , s) = 1 . In this paper, some relations between cs(G) and | Cent (G) | are established in the case where cs (G) = { 1 , p n , p n − 1 s } . Further, when p ∈ { 2 , 3 } , we determine the values of s and the structure of a finite group G such that cs (G) = { 1 , p n , p n − 1 s } . We also describe the structure of an AC-group G such that [ G : Z (G) ] = 3 n s and | Cent (G) | = 1 + ∑ i = 0 n 3 i , where gcd (3 , s) = 1 . [ABSTRACT FROM AUTHOR]

We construct 2-step nilpotent Lie algebras using labeled directed simple graphs and give a criterion to detect certain ideals and subalgebras by finding special subgraphs. We prove that if a label occurs only once, then reversing its orientation leads to an isomorphic algebra. As a consequence, if every edge is labeled differently, the Lie algebra depends only on the underlying undirected graph. In addition, we construct the graphs of all 2-step nilpotent Lie algebras of dimension ≤ 6 and compute the algebra of strata preserving derivations of the Lie algebra associated with the complete bipartite graph K m , n with two different labelings. [ABSTRACT FROM AUTHOR]

Let S be a commutative semigroup with zero. The zero-divisor graph associated with S is the simple graph whose vertex set is given by the nonzero zero-divisors of S where two distinct vertices are adjacent if and only if their product is zero. A graph is called complemented if every vertex has an incident edge which is not the edge of any three cycle. Complemented zero-divisor graphs which are associated with a finite commutative semigroup are described in terms of their clique number. The algebraic properties of the finite semigroups S whose zero-divisor graphs are complemented are described in terms of the clique number of the graph. Infinite families of complemented zero-divisor graphs associated with a finite commutative semigroup are given. [ABSTRACT FROM AUTHOR]

In this paper, we unfold balanced and totally balanced neighborhood hypergraphs to discover new classes of normally torsion-free monomial ideals. As a consequence, we establish that the closed neighborhood ideals and the dominating ideals of strongly chordal graphs are normally torsion-free. We discuss the stable sets of associated primes of the dominating ideals of cycles and characterize all the cycles with normally torsion-free dominating ideals. [ABSTRACT FROM AUTHOR]

Let R be a ring with unity and M a left R-module. We investigate a new graph associated with M called the simple-intersection graph of M, denoted by G S R (M) . Our focus is on exploring the relationship between different algebraic properties of M and properties G S R (M) , including connectedness, girth and dominating sets. Our results encompass determining the girth and diameter of GSR(M) and providing insights on the clique and domination numbers. [ABSTRACT FROM AUTHOR]

Let I = I (D) be the edge ideal of a weighted oriented graph D, let G be the underlying graph of D, and let I (n) be the n-th symbolic power of I defined using the minimal primes of I. We prove that I 2 = I (2) if and only if the following conditions hold: (i) every vertex of D with weight greater than 1 is a sink and (ii) G has no triangles. Using a result of Mandal and Pradhan and the classification of normally torsion-free edge ideals of graphs, we prove that I n = I (n) for all n ≥ 1 if and only if the following conditions hold: (a) every vertex of D with weight greater than 1 is a sink and (b) G is bipartite. If I has no embedded primes, conditions (a) and (b) classify when I is normally torsion-free. Using polyhedral geometry and integral closure, we give necessary conditions for the equality of ordinary and symbolic powers of monomial ideals with a minimal irreducible decomposition. Then, we classify when the dual of the edge ideal of a weighted oriented graph is normally torsion-free. [ABSTRACT FROM AUTHOR]

For an edge ideal I(G) of a simple graph G , we study the N -graded Betti numbers that appear in the linear strand of the minimal free resolution of I (Γ (Z n)) , where Γ (Z n) is the comaximal graph of the integral modulo ring Z n . We show that the extremal Betti number of I (Γ (Z n)) is ϕ (n) , where ϕ (n) is the Euler's totient function and thereby we obtain a large class of edge ideals with even extremal Betti numbers. We find the regularity (Castelnuovo-Mumford) and the projective dimension of these ideals. Moreover, we exhibit explicit formulae that determine all the N -graded Betti numbers in the linear strand of the minimal free resolution of I (Γ (Z n)) for certain values of n. [ABSTRACT FROM AUTHOR]

Let Δ be an (abstract) simplicial complex on n vertices. One can define the Artinian monomial algebra A (Δ) = k [ x 1 , ... , x n ] / 〈 x 1 2 , ... , x n 2 , I Δ 〉 , where k is a field of characteristic 0 and I Δ is the Stanley-Reisner ideal associated to Δ. In this paper, we aim to characterize the Weak Lefschetz Property (WLP) of A (Δ) in terms of the simplicial complex Δ. We are able to completely analyze when the WLP holds in degree 1, complementing work by Migliore, Nagel and Schenck on the WLP for quotients by quadratic monomials. We give a complete characterization of all 2-dimensional pseudomanifolds Δ such that A (Δ) satisfies the WLP. We also construct Artinian Gorenstein algebras that fail the WLP by combining our results and the standard technique of Nagata idealization. [ABSTRACT FROM AUTHOR]

Let G be a permutation group on a set Ω. A subset of Ω is a base for G if its pointwise stabilizer in G is trivial. By b(G) we denote the size of the smallest base of G. Every permutation group with b(G) = 2 contains some regular suborbits. It is conjectured by Burness-Giudici that every primitive permutation group G with b(G) = 2 has the property that if α g ∉ Γ then Γ ∩ Γ g ≠ ∅ , where Γ is the union of all regular suborbits of G relative to α. An affirmative answer of the conjecture has been shown for many sporadic simple groups and some alternating groups, but it is still open for simple groups of Lie-type. The first candidate of an infinite family of simple groups of Lie-type we should work on might be PSL (2 , q) , where q ≥ 5 . In this manuscript, we show the correctness of the conjecture for all the primitive groups with socle PSL (2 , q) , see Theorem 1.3. [ABSTRACT FROM AUTHOR]

Abstract–We prove a few results about non-nilpotent graphs of symmetric groups Sn – namely that they satisfy a conjecture of Nongsiang and Saikia (which is likewise proved for alternating groups An), and that for n ⩾ 19 each vertex has degree at least n ! 2 . We also show that the class of non-nilpotent graphs does not have any "local" properties, i.e. for every simple graph X there is a group G, such that its non-nilpotent graph contains X as an induced subgraph. [ABSTRACT FROM AUTHOR]

In this paper, we show that the cover ideals of theta graphs are normal square-free monomial ideals, and also satisfy the strong persistence property. Moreover, we re-prove a well-known result which is related to the cover ideals of bipartite graphs. We finally terminate this paper by stating some theorems which introduce some classes of normal monomial ideals. [ABSTRACT FROM AUTHOR]

A graphical regular representation of a group G is a Cayley graph of G whose full automorphism group is equal to the right regular permutation representation of G. In this paper, we prove that for Ree groups Ree (q) with q > 3, with probability tending to 1 as q → ∞ , a random involution y together with a fixed element x with order q – 1 gives rise to a cubic graphical regular representation of Ree (q) . A similar result involving a fixed element with order q + 3 q + 1 is also proved with the help of certain properties of Ree (q) given in [Leemans, D. Liebeck, M. W. (2017). Chiral polyhedra and finite simple groups. Bull. London Math. Soc. 49: 581–592]. [ABSTRACT FROM AUTHOR]

For a finite group G, and a subset S of G (possibly, contains the identity element), the bi-Cayley graph Bcay(G, S) of G with respect to S is defined as the undirected bipartite graph with vertex set G × { 0 , 1 } and edge set { { (g , 0) , (g s , 1) } | g ∈ G , s ∈ S } . In this paper, we give some sufficient conditions for the existence of hamiltonian cycle in Bcay (G = Z m ⋊ H , S) , where G = Z m ⋊ H is a semiproduct of Zm by a subgroup H of G. In addition, we introduce a new graph operation, which produces some classes of bi-Cayley graphs having hamiltonian cycle. [ABSTRACT FROM AUTHOR]

In this this paper all binary codes of length 552 which admit the sporadic simple group Co 3 as an imprimitive transitive permutation group are determined. Our aim is to understand the results also by using theoretical arguments and to discuss the combinatorial properties of the codes as well as their relation to some special properties of the Leech lattice group Co 3 . For all codes (with two exceptions) we obtain the weight enumerators and in many interesting cases the classification of codewords under the action of the group of code automorphisms Co 3 . The exceptional codes are both self-dual and have minimum weight 12. [ABSTRACT FROM AUTHOR]

For a group G, we define the (directed and undirected) prime index elements graph of G. We study some interplay between the algebraic properties of a group and the graph theoretic properties of its (directed and undirected) prime index elements graphs. Also we describe some relationships of this graph with the power graph and the reduced power graph of a group. [ABSTRACT FROM AUTHOR]

In this paper we consider the monoid D P S n of all partial isometries of a star graph Sn with n vertices. Our main objectives are to determine the rank and to exhibit a presentation of D P S n . We also describe Green's relations of D P S n and calculate its cardinal. [ABSTRACT FROM AUTHOR]

The commuting graph of a finite group is the undirected graph whose vertex set is the set of all elements of this group, and two distinct vertices are adjacent if they commute. In this paper, we characterize the strong metric dimension of the commuting graph of a finite group and give upper and lower bounds for the metric dimension of the commuting graph of a finite group. As applications, we compute the metric and strong metric dimension of the commuting graph of a dihedral group, a generalized quaternion group and a semidihedral group. [ABSTRACT FROM AUTHOR]

The power graph P (G) of a group G is the simple graph with vertex set G and two distinct vertices are adjacent if one of them is a positive power of the other. For a finite noncyclic nilpotent group G, we study the minimum degree δ (P (G)) of P (G). Under some conditions involving the prime divisors of | G | and the Sylow subgroups of G, we identify certain vertices associated with the generators of maximal cyclic subgroups of G such that δ (P (G)) is the degree of one of them. As an application, we obtain δ (P (G)) for some classes of nilpotent groups G. [ABSTRACT FROM AUTHOR]

A plenty of contributions have been done on symmetric Cayley graphs on nonabelian simple groups, but the only known complete classification of such graphs with composite valency is of valency 4 (provided 2-arc-transitivity) by Fang et al. [Europ. J. Combin. 25 (2004), 1107–1116] and Du and Feng [Comm. Algebra 47 (2019), 4565–4574]. This naturally motivates this work for classifying 2-arc-transitive hexavalent Cayley graphs on nonabelian simple groups. It is proved that these graphs are either normal or (A n , 2) -arc-transitive Cayley graphs on A n − 1 where n is among 11 specific numbers dividing 2 7 · 3 3 · 5 3. A specific example is also constructed. [ABSTRACT FROM AUTHOR]

This paper deals with the classification of groups G such that power graphs and proper power graphs of G are line graphs. In fact, we classify all finite nilpotent groups whose power graphs are line graphs. Also, we categorize all finite nilpotent groups (except non-abelian 2-groups) whose proper power graphs are line graphs. Moreover, we investigate when the proper power graphs of generalized quaternion groups are line graphs. Besides, we derive a condition on the order of the dihedral groups for which the proper power graphs of the dihedral groups are line graphs. [ABSTRACT FROM AUTHOR]

In this paper, we classify symmetric basic prime-valent graphs of order four times an odd prime power. As an application, prime-valent symmetric graphs of order four times a prime or prime square are specifically determined. We remark that special cases for valency 5 and 7 were characterized by Yang etc. (Eur. J. Combin. 80:236–246, 2019) and Pan and Yin (J. Algebr. Appl. 17(5):1850093, 2018). [ABSTRACT FROM AUTHOR]

Let Γ be a finite simple graph. If for some integer n ≥ 4 , Γ is a Kn-free graph whose complement has an odd cycle of length at least 2 n − 5 , then we say that Γ is an n-exact graph. For a finite group G, let Δ (G) denote the character graph built on the set of degrees of the irreducible complex characters of G. In this paper, we prove that the order of an n-exact character graph is at most 2 n − 1. Also we determine the structure of all finite groups G with extremal n-exact character graph. [ABSTRACT FROM AUTHOR]

A Cayley (di)graph Cay (G , S) of a group G is called normal if the right regular representation of G is normal in the full automorphism group of Cay (G , S) , and a CI-(di)graph if for every Cayley (di)graph Cay (G , T) , Cay (G , S) ≅ Cay (G , T) implies that there is σ ∈ Aut (G) such that S σ = T. We call a group G an NDCI-group or NCI-group if all normal Cayley digraphs or graphs of G are CI-digraphs or CI-graphs, respectively. We prove that a cyclic group of order n is an NDCI-group if and only if 8 ∤ n , and an NCI-group if and only if either n = 8 or 8 ∤ n. [ABSTRACT FROM AUTHOR]

Levelness and almost Gorensteinness are well-studied properties on graded rings as a generalized notion of Gorensteinness. In the present paper, we study those properties for the edge rings of the complete multipartite graphs, denoted by k [ K r 1 , ... , r n ] with 1 ≤ r 1 ≤ ⋯ ≤ r n. We give the complete characterization of which k [ K r 1 , ... , r n ] is level in terms of n and r 1 , ... , r n. Similarly, we also give the complete characterization of which k [ K r 1 , ... , r n ] is almost Gorenstein in terms of n and r 1 , ... , r n. [ABSTRACT FROM AUTHOR]

The orthogonality graph of a ring R, written as O(R), is an undirected simple graph with vertex set Z # (R) , the set of all nonzero two-sides zero-divisors of R, and for distinct x , y ∈ Z # (R) there exists an edge joining x and y if and only if x y = y x = 0. In this paper, we describe the automorphism group of O(R) when R is a finite semisimple ring. [ABSTRACT FROM AUTHOR]

The power graph P(G) of a group G is a graph with vertex set G, where two vertices u and v are adjacent if and only if u ≠ v and u m = v or v m = u for some positive integer m. In this paper, we raise and study the following question: For which natural numbers n every two groups of order n with isomorphic power graphs are isomorphic? In particular, it is proved that all such odd number n are cube-free and also they are not multiples of 16 in general. Moreover, we show that if two finite groups have isomorphic power graphs and one of them is nilpotent, the same is true for the other one. [ABSTRACT FROM AUTHOR]

Let G be a finite group and let Irr (G) be the set of all irreducible complex characters of G. For a character χ ∈ Irr (G) , the number cod (χ) : = | G : ker (χ) | / χ (1) is called the co-degree of χ. The set of co-degrees of all irreducible characters of G is denoted by cod (G). In this paper, we show that for a nontrivial finite group G, | Irr (G) | = | cod (G) | if and only if G is isomorphic to Z 2 or S3. [ABSTRACT FROM AUTHOR]

In recent years, the word magma has been used to designate a pair of the form (S , ∗) where * is a binary operation on the set S. Inspired by that terminology, we use the notation and terminology M (S) (the magma of S) to denote the set of all binary operations on the set S (i.e. the set of all magmas with underlying set S.) We study distributivity relations among magmas in the context of a hierarchy graph having M (S) as vertices and edges occurring precisely when an operation distributes over another one. The graph theoretic imagery and terminology serve to motivate questions in an intuitive way and to express their solutions in a reasonable fashion. While most considerations are done in general for arbitrary sets, particular emphasis is placed frequently on the case when S = n , the archetypal set with n elements. In that case, parameters such as cardinalities of outsets and insets, fully connected subsets, and longest cycle-free paths are explored. [ABSTRACT FROM AUTHOR]

We characterize colour-preserving automorphism vertex transitivity and vertex transitivity of the Cayley graphs of all semigroups in a class of pseudo-unitary homogeneous semigroups. [ABSTRACT FROM AUTHOR]

Let G be a finite group. The order supergraph of G is the graph with vertex set G, and two distinct vertices x, y are adjacent if o (x) | o (y) or o (y) | o (x). The enhanced power graph of G is the graph whose vertex set is G, and two distinct vertices are adjacent if they generate a cyclic subgroup. The reduced power graph of G is the graph with vertex set G, and two distinct vertices x, y are adjacent if 〈 x 〉 ⊂ 〈 y 〉 or 〈 y 〉 ⊂ 〈 x 〉. In this article, we characterize the strong metric dimension of the order supergraph, the enhanced power graph and the reduced power graph of a finite group. [ABSTRACT FROM AUTHOR]

For a finite group G, let R(G) be the solvable radical of G. The character-graph Δ (G) of G is a graph whose vertices are the primes which divide the degrees of some irreducible complex characters of G and two distinct primes p and q are joined by an edge if the product pq divides some character degree of G. In this paper we prove that, if Δ (G) has no subgraph isomorphic to K 3 , 3 and it's complement is non-bipartite, then G R (G) is an almost simple group with socle isomorphic to PSL 2 (q) where q ⩾ 5 is a prime power. Also we study the structure of all planar graphs that occur as the character-graph Δ (G) of a finite group G. [ABSTRACT FROM AUTHOR]

Suppose Γ is a finite simple graph. If D is a dominating set of Γ such that each x ∈ D is contained in the set of vertices of an odd cycle of Γ, then we say that D is an odd dominating set for Γ. For a finite group G, let Δ (G) denote the character graph built on the set of degrees of the irreducible complex characters of G. In this paper, we show that the complement of Δ (G) contains an odd dominating set, if and only if Δ (G) is a disconnected graph with non-bipartite complement. [ABSTRACT FROM AUTHOR]

A finite group G is called a DCI -group if every two isomorphic Cayley digraphs over G are Cayley isomorphic, i.e. their connection sets are conjugate by a group automorphism. We prove that the group C 4 × C p 2 , where p is a prime, is a DCI -group if and only if p ≠ 2. [ABSTRACT FROM AUTHOR]

The enhanced power graph P e (G) of a group G is a graph with vertex set G and two vertices are adjacent if they belong to the same cyclic subgroup. In this paper, we consider the minimum degree, independence number, and matching number of enhanced power graphs of finite groups. We first study these graph invariants for P e (G) when G is any finite group and then determine them when G is a finite abelian p-group, U 6 n = 〈 a , b : a 2 n = b 3 = e , b a = a b − 1 〉 , the dihedral group D 2 n , or the semidihedral group S D 8 n . If G is any of these groups, we prove that P e (G) is perfect and then obtain its strong metric dimension. Additionally, we give an expression for the independence number of P e (G) for any finite abelian group G. These results along with certain known equalities yield the edge connectivity, vertex covering number, and edge covering number of enhanced power graphs of the respective groups as well. [ABSTRACT FROM AUTHOR]

We study minimal cut-sets of the power graph of a finite non-cyclic nilpotent group which are associated with its maximal cyclic subgroups. As a result, we find the vertex connectivity of the power graphs of certain finite non-cyclic abelian groups whose order is divisible by at most three distinct primes. [ABSTRACT FROM AUTHOR]

In this article, we describe the Cayley graphs of pseudo-unitary homogeneous semigroups with zero 0, which are 0-disjoint unions of their subsets indexed by a groupoid. [ABSTRACT FROM AUTHOR]

A quasi-equigenerated monomial ideal I in the polynomial ring R = k [ x 1 , ... , x n ] is a Freiman ideal if μ (I 2) = l (I) μ (I) − ( l (I) 2 ) where l(I) is the analytic spread of I and μ (I) is the number of minimal generators of I. Freiman ideals are special since there exists an exact formula computing the minimal number of generators of any of their powers. In this work, we address the question of characterizing which cover ideals of simple graphs are Freiman. [ABSTRACT FROM AUTHOR]

Let Γ be a graph with vertex set V (Γ). A subset C of V (Γ) is a perfect code of Γ if C is an independent set such that every vertex in V (Γ) ∖ C is adjacent to exactly one vertex in C. A subset T of V (Γ) is a total perfect code of Γ if every vertex of Γ is adjacent to exactly one vertex in T. Let G be a finite group. The proper reduced power graph of G is the undirected graph whose vertex set consists of all nonidentity elements, and two distinct vertices x and y are adjacent if 〈 x 〉 ⊂ 〈 y 〉 or 〈 y 〉 ⊂ 〈 x 〉. In this paper, we give a necessary and sufficient condition for a proper reduced power graph to contain a perfect code. In particular, we determine all perfect codes of a proper reduced power graph provided that the proper reduced power graph admits a perfect code. Moreover, we give some necessary conditions for a proper reduced power graph to contain a total perfect code. As applications, we determine the abelian groups and generalized quaternion groups whose proper reduced power graphs admit a total perfect code. We also characterize all finite groups whose proper reduced power graphs admit a total perfect code of size 2. [ABSTRACT FROM AUTHOR]