INTERSECTION GRAPH OF SUBMODULES OF A MODULE

Let [Formula: see text] be a ring with unity. The cozero-divisor graph of a ring [Formula: see text] is an undirected simple graph whose vertices are the set of all nonzero and non-unit elements of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] and [Formula: see text]. The reduced cozero-divisor graph of a ring [Formula: see text] is an undirected simple graph whose vertex set is the set of all nontrivial principal ideals of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] and [Formula: see text]. Jesili et al. [On the genus of reduced cozero-divisor graph of commutative rings, Soft Comput. 27(2) (2023) 657–666] classified all non-local commutative rings whose reduced cozero-divisor graph is of genus one. In this paper, we characterize all classes of finite non-local commutative rings whose reduced cozero-divisor graph is of genus two. Moreover, we classify all the non-local commutative rings whose cozero-divisor graph is of genus two.

Let be a commutative ring with , and be an -module. The minimal intersection graph of , denoted by is a simple undirected graph whose vertices are proper non-zero submodules of and any two distinct vertices and are adjacent if and only if be an minimal (= simple) submodule of . In this article, we explore connectedness, clique number, split character, planarity, independence number and domination number of .

Let G be a finite abelian group with identity 0. For an integer the additive power graph of G is the simple undirected graph with vertex set G in which two distinct vertices x and y are adjacent if and only if x + y = nt for some with When the additive power graph has been studied in the name of square graph of finite abelian groups. In this paper, we study the additive power graph of G with n = 3 and name the graph as the cubic power graph. The cubic power graph of G is denoted by More specifically, we obtain the diameter and the girth of the graph and its complement Using these, we obtain a condition for and its complement to be self-centered. Also, we obtain the independence number, the clique number and the chromatic number of and its complement and hence we prove that and its complement are weakly perfect. Also, we discuss about the perfectness of At last, we obtain a condition for and its complement to be vertex pancyclic.

Let $G=(V, E)$ be a simple undirected graph. For a given set $L\subset \mathbb{R}$, a function $\omega: E \longrightarrow L$ is called an $L$-flow. Given a vector $\gamma \in \mathbb{R}^V$, $\omega$ is a $\gamma$-$L$-flow if for each $v\in V$, the sum of the values on the edges incident to $v$ is $\gamma(v)$. If $\gamma(v)=c$, for all $v\in V$, then the $\gamma$-$L$-flow is called a $c$-sum $L$-flow. In this paper, the existence of $\gamma$-$L$-flows for various choices of sets $L$ of real numbers is studied, with an emphasis on 1-sum flows. Let $L$ be a subset of real numbers containing $0$ and denote $L^*:=L\setminus \{0\}$. Answering a question from S. Akbari, M. Kano, and S. Zare. A generalization of $0$-sum flows in graphs. \emph{Linear Algebra Appl.}, 438:3629--3634, 2013.], the bipartite graphs which admit a $1$-sum $\mathbb{R}^*$-flow or a $1$-sum $\mathbb{Z}^*$-flow are characterized. It is also shown that every $k$-regular graph, with $k$ either odd or congruent to 2 modulo 4, admits a $1$-sum $\{-1, 0, 1\}$-flow.

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] be the set of all nonzero zero-divisors of [Formula: see text]. The annihilator graph of commutative ring [Formula: see text] is the simple undirected graph AG([Formula: see text]) with vertices [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] The essential graph of [Formula: see text] is defined as the graph [Formula: see text] with the vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] is an essential ideal. In this paper, we classify all finite commutative rings with identity whose annihilator graph and essential graph have crosscap two.

Let R be a ring with unity and I(R)∗ are all non-trivial left ideals of R. The intersection graph of ideals of R is denoted by G(R) is an undirected simple graph with vertex set I(R)∗ and two distinct vertices I and J are adjacent if and only if I∩J ≠ 0. In this article, we investigate some basic properties of the line graph associated to G(R), denoted by L(G(R)). Moreover, we investigate completeness, unicyclicness, bipartiteness, planarity, outerplanarity, ring graph, diameter, girth and clique of L(G(Zn)). We also investigate some basic properties of L(G(R)) for left Artinian ring and finally, we determine the domination number and bondage number of L(G(Zn)).

Let R be a commutative ring with non-zero identity and Z^∗(R) be the set of non-zero zero-divisors of R. The zero-divisor graph of R, denoted by Γ(R), is a simple undirected graph with all non-zero zero-divisors as vertices and two distinct vertices x, y ∈ Z^∗(R) are adjacent if and only if xy = 0. In this paper, the eigenvalues of Γ(Zn) for n = p^2q^2, where p and q are distinct primes, are investigated. Also, the girth, diameter, clique number and stability number of this graph are found.

For any nonempty set Ω and k-subset Λ, the k-intersection graph, denoted by Γm(Ω,Λ), is an undirected simple graph whose vertices are all m-subsets of Ω and two distinct vertices A and B are adjacent if and only if A∩B ⊈ Λ. In this paper, we determine diameter, girth, some numerical invariants and planarity, Hamiltonian and perfect matching of these graphs. finally we investigate their adjacency matrices.

لتكن حلقة ابدالية أحادية، وليكن مقاساً ايسر احادي. بيان تقاطع صغير من النمط-g للمقاس يرمز له , هو بيان غير مباشر بسيط رؤوسه تقابل كل المقاسات الجزئية غير التافهة في وكل رأسين مختلفين متجاورين أذا وفقط أذا كان التقاطع بينهما هو صغير من النمط-g. في هذا المقال، تم دراسة العلاقة بين الخصائص الجبرية الى والخصائص البيانية الى . أعتبر خصائص مثل الترابط، الكمال. علاوة على ذلك، يتم تحدد القطر و الطوق الى ، وكذلك أعطى صيغة لحساب الزمرة الى جانب أعداد الهيمنة الى .

Given a finite group G, the co-prime order graph of G is the simple undirected graph whose vertex set is G, and two distinct vertices x; y are adjacent if gcd(o(x); o(y)) = 1 or p, where p is a prime, and o(x) and o(y) are the orders of x and y, respectively. In this paper, we prove that, for a fixed positive integer k, there are finitely many finite groups whose co-prime order graphs have (non)orientable genus k. As applications, we classify all finite groups whose co-prime order graphs have (non)orientable genus one and two.

A subset S of V (G), where G is a simple undirected graph, is hop dominating if for each v ∈ V (G) \ S, there exists w ∈ S such that dG(v, w) = 2 and it is hop differentiating if N2 G[u] ∩ S ̸= N2 G[v] ∩ S for any two distinct vertices u, v ∈ V (G). A set S ⊆ V (G) is hop differentiating hop dominating if it is both hop differentiating and hop dominating in G. The minimum cardinality of a hop differentiating hop dominating set in G, denoted by γdh(G), is called the hop differentiating hop domination number of G. In this paper, we investigate some properties of this newly defined parameter. In particular, we characterize the hop differentiating hop dominating sets in graphs under some binary operations.

Let R be a ring and , where and are the sets of all left and right zero-divisors of R, respectively. The zero-divisor graph of R, denoted by , is a simple undirected graph with vertex set , and two distinct vertices are adjacent if and only if ab = 0 or ba = 0. Let , the ring of integers modulo n, and the ring of all matrices over a finite field of q elements. In this article, using the technique on characteristic matrices, we give the value of fixing number of . Moreover, we calculate the fixing number and metric dimension of and .

In this paper, a new kind of graph on a finite group , namely the result involution graph is defined and studied. We use to denote this graph, is a simple undirected graph with vertex set. Two distinct vertices are adjacent if and only if their product is nontrivial involution element in . The result involution graph for several finite groups are obtained. We study some properties of the result involution graph by resizing graph by using the conjugacy classes of . Finally, we show that the result involution graphs for the symmetric groups and the alternating groups are connected with diameter at most 3 and radius at most 2 for. Furthermore, they have girth 3.

A simple undirected graph is said to be semisymmetric if it is regular and edge-transitive but not vertex-transitive. Every semisymmetric graph is a bipartite graph with two parts of equal size. Let p be a prime. In this paper, a class of semisymmetric graphs of order 2 p 3 are determined. This work is a partial result for our long term goal to classify all semisymmetric graphs of order 2 p 3 .

A simple undirected graph is said to be semisymmetric if it is regular and edge-transitive, but not vertex-transitive. Every semisymmetric graph is a bipartite graph with two biparts of equal size. It was proved by Folkman in (J Comb Theory Ser B 3:215–232, 1967) that there exist no semisymmetric graphs of order $$2p$$ and $$2p^2$$ , where $$p$$ is a prime. For any distinct primes $$p$$ and $$q$$ , the classification of semisymmetric graphs of order $$2pq$$ was given by Du and Xu in (Comm Algebra 28:2685–2715, 2000). Naturally, one of our long-term goals is to determine all the semisymmetric graphs of order $$2p^3$$ , for any prime $$p$$ . All these graphs $$\Gamma $$ are divided into two subclasses: (I) the automorphism group $$\hbox {Aut }(\Gamma )$$ acts unfaithfully on at least one bipart; and (II) $$\hbox {Aut }(\Gamma )$$ acts faithfully on both biparts. In Wang and Du (Eur J Comb 36:393–405, 2014), a group theoretical characterization for Subclass (I) was given by the authors. Based on this characterization, this paper gives a complete classification for Subclass (I).