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Let G be a finite group and construct a graph Δ(G) by taking G {1} as the vertex set of Δ(G) and by drawing an edge between two vertices x and y if 〈x, y〉 is cyclic. Let K(G) be the set consisting of the universal vertices of Δ(G) along the identity element. For a solvable group G, we present a necessary and sufficient condition for K(G) to be nontrivial. We also develop a connection between Δ(G) and K(G) when ∣G∣ is divisible by two distinct primes and the diameter of Δ(G) is 2. [ABSTRACT FROM AUTHOR]

The co-maximal subgroup graph Γ (G) of a group G is a graph whose vertices are non-trivial proper subgroups of G and two vertices H and K are adjacent if H K = G . In this paper, we continue the study of Γ (G) , especially when Γ (G) has isolated vertices. We define a new graph Γ ∗ (G) , which is obtained by removing isolated vertices from Γ (G) . We characterize when Γ ∗ (G) is connected, a complete graph, star graph, has an universal vertex etc. We also find various graph parameters like diameter, girth, bipartiteness etc. in terms of properties of G. [ABSTRACT FROM AUTHOR]

V. Levenshtein first proposed the sequence reconstruction problem in 2001. This problem studies the same sequence from some set is transmitted over multiple channels, and the decoder receives the different outputs. Assume that the transmitted sequence is at distance d from some code and there are at most r errors in every channel. Then the sequence reconstruction problem is to find the minimum number of channels required to recover exactly the transmitted sequence that has to be greater than the maximum intersection between two metric balls of radius r, where the distance between their centers is at least d. In this paper, we study the sequence reconstruction problem of permutations under the Hamming distance. In this model we define a Cayley graph over the symmetric group, study its properties and find the exact value of the largest intersection of its two metric balls for d = 2 r . Moreover, we give a lower bound on the largest intersection of two metric balls for d = 2 r - 1 . [ABSTRACT FROM AUTHOR]

Let n = 2 s p 1 α 1 ⋯ p k α k , where s = 0 or 1, α i ≥ 1 , and the distinct primes p i satisfy p i ≡ 1 (mod 4) for all i = 1 , … , k . Let Z n ∗ denote the group of units in the commutative ring Z n . In a recent paper, we defined the Paley-type graph G n of order n as the graph whose vertex set is Z n and xy is an edge if x - y ≡ a 2 (mod n) for some a ∈ Z n ∗ . Computing the number of cliques of a particular order in a Paley graph or its generalizations has been of considerable interest. In our recent paper, for primes p ≡ 1 (mod 4) and α ≥ 1 , by evaluating certain character sums, we found the number of cliques of order 3 in G p α and expressed the number of cliques of order 4 in G p α in terms of Jacobi sums. In this article we give combinatorial proofs and find the number of cliques of orders 3 and 4 in G n for all n for which the graph is defined. [ABSTRACT FROM AUTHOR]

A Cayley digraph Cay (G , S) of a group G with respect to a subset S of G is called a CI-digraph if for every Cayley digraph Cay (G , T) isomorphic to Cay (G , S) , there exists an α ∈ Aut (G) such that S α = T . For a positive integer m, G is said to have the m-DCI property if all Cayley digraphs of G with out-valency m are CI-digraphs. Li (European J Combin 18:655–665, 1997) gave a necessary condition for cyclic groups to have the m-DCI property, and in this paper, we find a necessary condition for dihedral groups to have the m-DCI property. Let D 2 n be the dihedral group of order 2n, and assume that D 2 n has the m-DCI property for some 1 ≤ m ≤ n - 1 . It is shown that n is odd, and if further p + 1 ≤ m ≤ n - 1 for an odd prime divisor p of n, then p 2 ∤ n . Furthermore, if n is a power of a prime q, then D 2 n has the m-DCI property if and only if either n = q , or q is odd and 1 ≤ m ≤ q . [ABSTRACT FROM AUTHOR]

Let R be a finite commutative ring with unity, and let G = (V , E) be a simple graph. The zero-divisor graph, denoted by Γ (R) is a simple graph with vertex set as R, and two vertices x , y ∈ R are adjacent in Γ (R) if and only if x y = 0 . In [5], the authors have studied the Laplacian eigenvalues of the graph Γ (Z n) and for distinct proper divisors d 1 , d 2 , ⋯ , d k of n, they defined the sets as, A d i = { x ∈ Z n : (x , n) = d i } , where (x, n) denotes the greatest common divisor of x and n. In this paper, we show that the sets A d i , 1 ≤ i ≤ k are actually orbits of the group action: A u t (Γ (R)) × R ⟶ R , where A u t (Γ (R)) denotes the automorphism group of Γ (R) . Our main objective is to determine new classes of threshold graphs, since these graphs play an important role in several applied areas. For a reduced ring R, we prove that Γ (R) is a connected threshold graph if and only if R ≅ F q or R ≅ F 2 × F q . We provide classes of threshold graphs realized by some classes of local rings. Finally, we characterize all finite commutative rings with unity of which zero-divisor graphs are not threshold. [ABSTRACT FROM AUTHOR]

For a commutative ring R , even elements in R as vertices and two distinct vertices x i , y j ∈ R are adjacent iff x i y j = 0 or y j x i = 0 , then the graph is known as an even prime graph. In compressed even prime graph, whose vertex set is the set of all equivalence classes of even elements in R , the equivalence classes elements are denoted by [ β ] or β ¯ and two distinct equivalence classes [ β i ] and [ β j ] are adjacent iff [ β i ] [ β j ] = 0 , graph is denoted by E P G E (Z n) . This paper delves into discussions regarding the chromatic number and clique number of such graphs across various families. Additionally, we discuss the chromatic number of the prime graph associated with the commutative ring R . [ABSTRACT FROM AUTHOR]

A skew morphism of a finite group A is a permutation φ of A fixing the identity element and for which there is an integer-valued function π on A such that φ (a b) = φ (a) φ π (a) (b) for all a , b ∈ A . A skew morphism φ of A is smooth if the associated power function π is constant on the orbits of φ , that is, π (φ (a)) ≡ π (a) (mod | φ |) for all a ∈ A . In this paper, we show that every skew morphism of a cyclic group of order n is smooth if and only if n = 2 e n 1 , where 0 ≤ e ≤ 4 and n 1 is an odd square-free number. A partial solution to a similar problem on non-cyclic abelian groups is also given. [ABSTRACT FROM AUTHOR]

We first determine the structure of the power digraphs of completely 0-simple semigroups, and then some properties of their power graphs are given. As the main result in this paper, using Cameron and Ghosh's theorem about power graphs of abelian groups, we obtain a characterization that two G 0 -normal completely 0-simple orthodox semigroups S and T with abelian group H -classes are isomorphic based on their power graphs. We also present an algorithm to determine that S and T are isomorphic or not. [ABSTRACT FROM AUTHOR]

Let Γ = (V , E) be a graph and a, b nonnegative integers. An (a, b)-regular set in Γ is a nonempty proper subset D of V such that every vertex in D has exactly a neighbours in D and every vertex in V \ D has exactly b neighbours in D. A (0, 1)-regular set is called a perfect code, an efficient dominating set, or an independent perfect dominating set. A subset D of a group G is called an (a, b)-regular set of G if it is an (a, b)-regular set in some Cayley graph of G, and an (a, b)-regular set in a Cayley graph of G is called a subgroup (a, b)-regular set if it is also a subgroup of G. In this paper, we study (a, b)-regular sets in Cayley graphs with a focus on (0, k)-regular sets, where k ≥ 1 is an integer. Among other things, we determine when a non-trivial proper normal subgroup of a group is a (0, k)-regular set of the group. We also determine all subgroup (0, k)-regular sets of dihedral groups and generalized quaternion groups. We obtain necessary and sufficient conditions for a hypercube or the Cartesian product of n copies of the cycle of length p to admit (0, k)-regular sets, where p is an odd prime. Our results generalize several known results from perfect codes to (0, k)-regular sets. [ABSTRACT FROM AUTHOR]

A bipartite graph Γ is a bi-Cayley graph over a group H if H ⩽ Aut Γ acts regularly on each part of Γ . A bi-Cayley graph Γ is said to be a normal bi-Cayley graph over H if H ⊴ Aut Γ , and bi-primitive if the bipartition preserving subgroup of Aut Γ acts primitively on each part of Γ . In this paper, a classification is given for 2-arc-transitive bi-Cayley graphs which are bi-primitive and non-normal. [ABSTRACT FROM AUTHOR]

The power graph of a group is the simple graph whose vertices are the group elements and two vertices are adjacent whenever one of them is a positive power of the other. In this paper, we characterize the finite nilpotent groups whose power graphs have equal vertex connectivity and minimum degree. [ABSTRACT FROM AUTHOR]

We continue the study of prime graphs of finite groups, also known as Gruenberg–Kegel graphs. The vertices of the prime graph of a finite group are the prime divisors of the group order, and two vertices p and q are connected by an edge if and only if there is an element of order pq in the group. Prime graphs of solvable groups have been characterized in graph theoretical terms only, as have been the prime graphs of groups whose only nonsolvable composition factor is A 5 . In this paper, we classify the prime graphs of all groups whose composition factors have arithmetically small orders, that is, have no more than three prime divisors in their orders. We find that all such graphs have 3-colorable complements, and we provide full characterizations of the prime graphs of such groups based on the exact type and multiplicity of the nonabelian composition factors of the group. [ABSTRACT FROM AUTHOR]

We propose a quantization algebra of the Loday-Ronco Hopf algebra k [ Y ∞ ] , based on the Topological Recursion formula of Eynard and Orantin. We have shown in previous works that the Loday-Ronco Hopf algebra of planar binary trees is a space of solutions for the genus 0 version of Topological Recursion, and that an extension of the Loday Ronco Hopf algebra as to include some new graphs with loops is the correct setting to find a solution space for arbitrary genus. Here we show that this new algebra k [ Y ∞ ] h is still a Hopf algebra that can be seen in some sense to be made precise in the text as a quantization of the Hopf algebra of planar binary trees, and that the solution space of Topological Recursion A TopRec h is a subalgebra of a quotient algebra A Reg h obtained from k [ Y ∞ ] h that nevertheless doesn't inherit the Hopf algebra structure. We end the paper with a discussion on the cohomology of A TopRec h in low degree. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the Harary index and the hyper-Wiener index of the H-generalized join of graphs and the generalized corona product of graphs. As a consequence, we deduce some of the results in (Das et al. in J. Inequal. Appl. 2013:339, 2013) and (Khalifeh et al. in Comput. Math. Appl. 56:1402–1407, 2008). Moreover, we calculate the Harary index and the hyper-Wiener index of the ideal-based zero-divisor graph of a ring. [ABSTRACT FROM AUTHOR]

Let G be a finite non-abelian group and Z(G) be its center. We associate a commuting graph Γ (G) to G, whose vertex set is G \ Z (G) and two distinct vertices are adjacent if they commute. In this paper we prove that the set of all non-abelian groups whose commuting graph has maximum vertex degree bounded above by a fixed k ∈ N is finite. Also, we characterize all groups for which the associated commuting graphs have maximum vertex degree at most 4. [ABSTRACT FROM AUTHOR]

The fact that the adjacency matrix of every finite graph is diagonalizable plays a fundamental role in spectral graph theory. Since this fact does not hold in general for digraphs, it is natural to ask whether it holds for digraphs with certain level of symmetry. Interest in this question dates back to the early 1980 s, when P. J. Cameron asked for the existence of arc-transitive digraphs with non-diagonalizable adjacency matrix. This was answered in the affirmative by Babai (J Graph Theory 9:363–370, 1985). Then Babai posed the open problems of constructing a 2-arc-transitive digraph and a vertex-primitive digraph whose adjacency matrices are not diagonalizable. In this paper, we solve Babai's problems by constructing an infinite family of s-arc-transitive digraphs for each integer s ≥ 2 , and an infinite family of vertex-primitive digraphs, both of whose adjacency matrices are non-diagonalizable. [ABSTRACT FROM AUTHOR]

Let p be a prime, and let Λ 2 p be a connected cubic arc-transitive graph of order 2p. In the literature, a lot of works have been done on the classification of edge-transitive normal covers of Λ 2 p for specific p ≤ 7 . An interesting problem is to generalize these results to an arbitrary prime p. In 2014, Zhou and Feng classified edge-transitive cyclic or dihedral normal covers of Λ 2 p for each prime p. In our previous work, we classified all edge-transitive N-normal covers of Λ 2 p , where p is a prime and N is a metacyclic 2-group. In this paper, we give a classification of edge-transitive N-normal covers of Λ 2 p , where p ≥ 5 is a prime and N is a metacyclic group of odd prime power order. [ABSTRACT FROM AUTHOR]

Consider a commutative ring with unity denoted as R , and let W (R) represent the set of non-unit elements in R . The coannihilator graph of R , denoted as A G ′ (R) , is a graph defined on the vertex set W (R) ∗ . This graph captures the relationships among non-unit elements. Specifically, two distinct vertices, x and y, are connected in A G ′ (R) if and only if either x ∉ x y R or y ∉ x y R , where w R denotes the principal ideal generated by w ∈ R . In the context of this paper, the primary objective is to systematically classify finite rings R based on distinct characteristics of their coannihilator graph. The focus is particularly on cases where the coannihilator graph exhibits a genus or crosscap of two. Additionally, the research endeavors to provide a comprehensive characterization of finite rings R for which the connihilator graph A G ′ (R) attains an outerplanarity index of two. [ABSTRACT FROM AUTHOR]

The common divisor graph, Γ (X) , is a graph that has been defined on a set of positive integers X. Some properties of this graph have been studied in the cases when either X is the set of character degrees or the set of conjugacy class sizes of a group. This paper deals with several properties of a particular case of the common divisor graph, Γ (Z) , when Z is a multiset of positive integers that admits a decomposition Z = X Y , where X Y = { x y | x ∈ X , y ∈ Y } and 1 ∈ X and 1 ∈ Y . Our results can be applied for the graphs associated to character degrees and conjugacy classes of the direct product of two finite groups. [ABSTRACT FROM AUTHOR]

Let G be a finite group generated by S and C(G, S) the Cayley digraphs of G with connection set S. In this paper, we give some sufficient conditions for the existence of hamiltonian circuit in C(G, S), where G = Zm ⋊ H is a semiproduct of Zm by a subgroup H of G. In particular, if m is a prime, then the Cayley digraph of G has a hamiltonian circuit unless G = Zm × H. In addition, we introduce a new digraph operation, called φ-semiproduct of Γ1 by Γ2 and denoted by Γ1 ⋊φ Γ2, in terms of mapping φ: V(Γ2) → {1, −1}. Furthermore we prove that C(Zm, {a}) ⋊φC(H, S) is also a Cayley digraph if φ is a homomorphism from H to { 1 , − 1 } ≤ Z m ∗ , which produces some classes of Cayley digraphs that have hamiltonian circuits. [ABSTRACT FROM AUTHOR]

In this paper, we extend our investigation about the generalized comaximal graph introduced in Biswas et al. (Discrete Math Algorithms Appl 11(1):1950013, 2019a). The generalized comaximal graph is defined as follows: given a finite commutative ring R, the generalized comaximal graph G(R) is an undirected graph with its vertex set comprising elements of R and two distinct vertices u, v are adjacent if and only if there exists a non-zero idempotent e ∈ R such that u R + v R = e R . In this study, we focus on identifying the rings R for which the graph G(R) exhibits planarity. Moreover, we provide a characterization of the class of ring for which G(R) is toroidal, denoted by γ (G (R)) = 1 . Furthermore, we also evaluate the energy of the graph G(R). Finally, we demonstrate that the graph G(R) is always Hamiltonian for any finite commutative ring R. [ABSTRACT FROM AUTHOR]

In this paper, we study the symmetry of the nonzero component graph Γ n (F) of an n-dimensional vector space over a field F . We explicitly compute the automorphism group of Γ n (F) and compute stabilizers, orbits, and the determining number of Γ n (F) . We then characterize all the determining sets of Γ n (F) . Using this characterization, we compute the determining polynomial of Γ n (F) when F is a finite field. We also discuss the core property and the Wiener index of Γ n (F) . [ABSTRACT FROM AUTHOR]

In this paper, we consider three submonoids of the dihedral inverse monoid D I n , namely its submonoids OPDI n , MDI n and ODI n of all orientation-preserving, monotone and order-preserving transformations, respectively. For each of these three monoids, we compute the cardinality, give descriptions of Green’s relations and determine the rank. [ABSTRACT FROM AUTHOR]

The enhanced power graph of a group G is a graph with vertex set G, where two distinct vertices x\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{x}$$\end{document} and y\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{y}$$\end{document} are adjacent if and only if there exists an element w\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{w}$$\end{document} in G such that both x\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{x}$$\end{document} and y\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{y}$$\end{document} are powers of w\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{w}$$\end{document}. To obtain the proper enhanced power graph, we consider the induced subgraph on the set G\D\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$G \setminus D$$\end{document}, where D represents the set of dominating vertices in the enhanced power graph. In this paper, we aim to determine the strong domination number of the proper enhanced power graphs of finite nilpotent groups. [ABSTRACT FROM AUTHOR]

An orientable vertex primitive complete map is a two-cell embedding of a complete graph into an orientable surface such that the automorphism group of this map acts primitively on its vertex set. The paper is devoted to the problem of enumerating orientable vertex primitive complete maps. For a given integer n, we derive the number of different such maps with n vertices. Furthermore, we obtain explicit formulas for the numbers of non-isomorphic orientable vertex primitive complete maps with n vertices. [ABSTRACT FROM AUTHOR]

The projective norm graphs are central objects to extremal combinatorics. They appear in a variety of contexts, most importantly they provide tight constructions for the Turán number of complete bipartite graphs K t , s with s > (t - 1) ! . In this note we deepen their understanding further by determining their automorphism group. [ABSTRACT FROM AUTHOR]

The relative fixity of a digraph Γ is defined as the ratio between the largest number of vertices fixed by a nontrivial automorphism of Γ and the number of vertices of Γ . We characterize the vertex-primitive digraphs whose relative fixity is at least 1 3 , and we show that there are only finitely many vertex-primitive digraphs of bounded out-valency and relative fixity exceeding a positive constant. [ABSTRACT FROM AUTHOR]

Let Δ be a connected arc-transitive G-graph which is locally finite and locally quasiprimitive. Let { x , y } be an edge of Δ . A relation between G x [ 1 ] / O p (G x [ 1 ]) and the existence of certain normal subgroups of G x Δ (x) and G x , y Δ (x) is established. This is then used to determine the vertex stabilizers of a class of 2-arc-transitive graphs with trivial edge kernel. [ABSTRACT FROM AUTHOR]

We consider 2-designs which admit a group of automorphisms that is flag-transitive and leaves invariant a chain of nontrivial point-partitions. We build on our recent work on 2-designs which are block-transitive but not necessarily flag-transitive. In particular we use the concept of the "array" of a point subset with respect to the chain of point-partitions; the array describes the distribution of the points in the subset among the classes of each partition. We obtain necessary and sufficient conditions on the array in order for the subset to be a block of such a design. By explicit construction we show that for any s ≥ 2 , there are infinitely many 2-designs admitting a flag-transitive group that preserves an invariant chain of point-partitions of length s. Moreover an exhaustive computer search, using Magma, seeking designs with e 1 e 2 e 3 points (where each e i ≤ 50 ) and a partition chain of length s = 3 , produced 57 such flag-transitive designs, among which only three designs arise from our construction—so there is still much to learn. [ABSTRACT FROM AUTHOR]

Given a graph X with a Hamilton cycle C, the compression factor κ(X,C)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\kappa (X,C)$$\end{document}of C is the order of the largest cyclic subgroup of Aut(C)∩Aut(X)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\textrm{Aut}}\,(C)\cap {\textrm{Aut}}\,(X)$$\end{document}, and the Hamilton compression κ(X)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\kappa (X)$$\end{document}of X is the maximum of κ(X,C)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\kappa (X,C)$$\end{document} where C runs over all Hamilton cycles in X. Generalizing the well-known open problem regarding the existence of vertex-transitive graphs without Hamilton paths/cycles, it was asked by Gregor et al. (Ann Comb, arXiv:2205.08126v1, https://doi.org/10.1007/s00026-023-00674-y, 2023) whether for every positive integer k, there exists infinitely many vertex-transitive graphs (Cayley graphs) with Hamilton compression equal to k. Since an infinite family of Cayley graphs with Hamilton compression equal to 1 was given there, the question is completely resolved in this paper in the case of Cayley graphs with a construction of Cayley graphs of semidirect products Zp⋊Zk\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {Z}_p\rtimes \mathbb {Z}_k$$\end{document} where p is a prime and k≥2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$k \ge 2$$\end{document} a divisor of p-1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p-1$$\end{document}. Further, infinite families of non-Cayley vertex-transitive graphs with Hamilton compression equal to 1 are given. All of these graphs being metacirculants, some additional results on Hamilton compression of metacirculants of specific orders are also given. [ABSTRACT FROM AUTHOR]

We prove that every regular graph of valency at least four whose automorphism group contains a nilpotent subgroup acting transitively on the vertex set admits a nowhere-zero 3-flow. [ABSTRACT FROM AUTHOR]

Suppose that Δ is a thick, locally finite and locally s-arc transitive G-graph with s ≥ 4 . For a vertex z in Δ , let G z be the stabilizer of z and G z [ 1 ] the kernel of the action of G z on the neighbours of z. We say Δ is of pushing up type provided there exist a prime p and a 1-arc (x, y) such that C G z (O p (G z [ 1 ])) ≤ O p (G z [ 1 ]) for z ∈ { x , y } and O p (G x [ 1 ]) ≤ O p (G y [ 1 ]) . We show that if Δ is of pushing up type, then O p (G x [ 1 ]) is elementary abelian and G x / G x [ 1 ] ≅ X with PSL 2 (p a) ≤ X ≤ P Γ L 2 (p a) . [ABSTRACT FROM AUTHOR]

Let R be a finite commutative ring with identity. We study the structure of the zero-divisor graph of R and then determine its vertex connectivity when: (i) R is a local principal ideal ring, and (ii) R is a finite direct product of local principal ideal rings. For such rings R, we also characterize the vertices of minimum degree and the minimum cut-sets of the zero-divisor graph of R. [ABSTRACT FROM AUTHOR]

A Coxeter group admits infinite-dimensional irreducible complex representations if and only if it is not finite or affine. In this paper, we provide a construction of some of those representations for certain Coxeter groups using some topological information of the corresponding Coxeter graphs. [ABSTRACT FROM AUTHOR]

We study two measures of associativity for graph algebras of finite undirected graphs: the index of nonassociativity and (a variant of) the semigroup distance. We determine “almost associative” and “antiassociative” graphs with respect to both measures. It turns out that the antiassociative graphs are exactly the balanced complete bipartite graphs, no matter which of the two measures we consider. In the class of connected graphs the two notions of almost associativity are also equivalent. [ABSTRACT FROM AUTHOR]

For a commutative ring R, the comaximal graph Γ (R) of R is a simple graph with vertex set R and two distinct vertices u and v of Γ (R) are adjacent if and only if a R + b R = R . In this article, we find the Laplacian eigenvalues of Γ (Z n) and show that the algebraic connectivity of Γ (Z n) is always an even integer and equals ϕ (n) , thereby giving a large family of graphs with integral algebraic connectivity. Further, we prove that the second largest Laplacian eigenvalue of Γ (Z n) is an integer if and only if n = p α q β , and hence Γ (Z n) is Laplacian integral if and only if n = p α q β , where p, q are primes and α , β are non-negative integers. This answers a problem posed by [Banerjee, Laplacian spectra of comaximal graphs of the ring Z n , Special Matrices, (2022)]. [ABSTRACT FROM AUTHOR]

A graph is a core or unretractive if all its endomorphisms are automorphisms. Well-known examples of cores include the Petersen graph and the graph of the dodecahedron—both generalized Petersen graphs. We characterize the generalized Petersen graphs that are cores. A simple characterization of endomorphism-transitive generalized Petersen graphs follows. This extends the characterization of vertex-transitive generalized Petersen graphs due to Frucht, Graver, and Watkins and solves a problem of Fan and Xie. Moreover, we study generalized Petersen graphs that are (underlying graphs of) Cayley graphs of monoids. We show that this is the case for the Petersen graph, answering a recent mathoverflow question, for the Desargues graphs, and for the Dodecahedron—answering a question of Knauer and Knauer. Moreover, we characterize the infinite family of generalized Petersen graphs that are Cayley graphs of a monoid with generating connection set of size two. This extends Nedela and Škoviera's characterization of generalized Petersen graphs that are group Cayley graphs and complements results of Hao, Gao, and Luo. [ABSTRACT FROM AUTHOR]

In this article, we provide an explicit description of a set of generators for any ideal of an ultragraph Leavitt path algebra. We provide several additional consequences of this description, including information about generating sets for graded ideals, the graded uniqueness and Cuntz-Krieger theorems, the semiprimeness, and the semiprimitivity of ultragraph Leavitt path algebras, a complete characterization of the prime and primitive ideals of an ultragraph Leavitt path algebra. We also show that every primitive ideal of an ultragraph Leavitt path algebra is exactly the annihilator of a Chen simple module. Consequently, we prove Exel's Effros-Hahn conjecture on primitive ideals in the ultragraph Leavitt path algebra setting (a conclusion that is also new in the context of Leavitt path algebras of graphs). [ABSTRACT FROM AUTHOR]