On graphs associated to topological spaces

A set of vertices and edges forms a graph. A graph can be associated with groups using the groups' properties for its vertices and edges. The set of vertices of the graph comprises the elements of the group, while the set of edges of the graph is the properties and requirements for the graph. A non-abelian tensor square graph of a group is defined when its vertex set represents the non-tensor centre elements' set of G. Then, two distinguished vertices are connected by an edge if and only if the non-abelian tensor square of these two elements is not equal to the identity of the non-abelian tensor square. This study investigates the non-abelian tensor square graph for the symmetric group of order six. In addition, some properties of this group's non-abelian tensor square graph are computed, including the diameter, the dominating number, and the chromatic number. The perfect code for the non-abelian tensor square graph for a symmetric group of order six is also found in this paper.

In topological spaces, which are branches of pure mathematics, are difficult to find, the significance of this paper is to find some in graph theory via some topological space concepts, we introduced some well-known concepts in topology and related them to a graph such as (graph interior, graph closure, graph exterior, graph boundary and graph limit point). We study a characterization and relationships between them and present some results and theorems about this notions. In this work includes all Species graphs.

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)).

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.

In this paper, we apply the notion of $e$-$\I$-open sets \cite{Wadei6} in ideal topological spaces to present and study new classes of functions called contra $e$-$\I$-continuous functions, almost-$e$-$\I$-continuous, almost contra-$e$-$\I$-continuous, and almost weakly-$e$-$\I$-continuous along with their several properties, characterizations and mutual relationships. Relationships between their new classes and other classes of functions are established and some characterizations of their new classes of functions are studied. Further, we introduce new types of graphs, called $e$-$\I$-closed, contra-$e$-$\I$-closed, and strongly contra-$e$-$\I$-closed graphs via $e$-$\I$-open sets. Several characterizations and properties of such notions are investigated.

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.

The relationship between edges and vertices is fundamental to graph theory, significantly influencing different graph properties and applications. The topological index is a numerical value calculable by specific techniques and graph properties. Meanwhile, the relative prime coprime graph of a group G for its subgroup H is a graph in which two distinct vertices are adjacent whenever the greatest common divisor of the order of both vertices is equal to one or relative prime and either or both are in H. In this paper, we introduce this definition and describe some properties. Moreover, we also define some reverse topological indices (reverse Harmonic, reverse Randic, reverse SK, reverse SK1, and reverse SK2 indices) for the relatively prime coprime graph. In particular, we describe these reverse degree-based topological indices for newly defined relatively prime coprime graphs based on the order of the elements of the integers modulo group.

The study of graph properties has gathered many attentions in the past years. The graph properties that are commonly studied include the chromatic number, the clique number and the domination number of a finite graph. In this study, a type of graph properties, which is the perfect code is studied. The perfect code is originally used in coding theory, then extended to other fields including graph theory. Hence, in this paper, the perfect code is determined for the commuting zero divisor graphs of some finite rings of matrices. First, the commuting zero divisor graph of the finite rings of matrices is constructed where its vertices are all zero divisors of the ring and two distinct vertices, say x and y, are adjacent if and only if xy = yx = 0. Then, from the vertex set of the graph, the neighborhood elements of the vertices are determined in order to compute the perfect codes of the graph.

Spectra of quasi-strongly regular graphs

Applications on color (distance) signless laplacian energy of annihilator monic prime graph of commutative rings

This paper considers two properties of graphs, one geometrical and one topological, and shows that they are strongly related. Let G be a graph with four distinguished and distinct vertices, w1 , w2 , b1 , b2 . Consider the two properties, TRI+ (G) and MONO(G), defined as follows. TRI+(G): There is a planar drawing of G such that all 3-cycles of G are faces; all faces of G are triangles except for the single face which is the 4-cycle (w1 -b1 -w2 -b2 -w1 ). MONO(G): G contains the 4-cycle (w1 -b1 -w2 -b2 -w1 ) and, for any labeling of the vertices of G by the colors {white, black} such that w1 and w2 are white, while b1 and b2 are black, precisely one of the following holds. There is a path of white vertices connecting w1 and w2 . There is a path of black vertices connecting b1 and b2 . Our main result is that a graph G enjoys property TRI+ (G) if and only if it is minimal with respect to property MONO. Building on this, we show that one can decide in polynomial time whether or not a given graph G has property MONO(G).

In the application of graph theory to problems arising in network design, the requirements of the network can be expressed in terms of restrictions on the values of certain graph parameters such as connectivity, edge‐connectivity, diameter, and independence number. In this paper, we focus on networks whose requirements translate into adjacency restrictions on the graph representing the network. More specifically, a graph G is said to have property P(m,n,k) if for any set of m + n distinct vertices there are at least k other vertices, each of which is adjacent to the first m vertices but not adjacent to any of the latter n vertices. The problem that arises is that of characterizing graphs having property P(m,n,k). In this paper, we present properties of graphs satisfying the adjacency property. In particular, for q  1(mod 4), a prime power, the Paley graph Gq of order q is the graph whose vertices are elements of the finite field 𝔽q; two vertices are adjacent if and only if their difference is a quadratic residue. For any m, n, and k, we show that all sufficiently large Paley graphs satisfy P(m,n,k). © 1993 by John Wiley & Sons, Inc.

In this paper, a new kind of graph on a commutative ring is introduced and investigated. Small intersection graph of a ring $R$, denoted by $G(R)$, is a graph with all non-small proper ideals of $R$ as vertices and two distinct vertices $I$ and $J$ are adjacent if and only if $I\cap J$ is not small in $R$. In this article, some interrelation between the graph theoretic properties of this graph and some algebraic properties of rings are studied. We investigated the basic properties of the small intersection graph as diameter, girth, clique number, cut vertex, planar property and independence number. Further, it is shown that the independence number of a small graph of a ring $R$ is equal to the number of its maximal ideals and the domination number of small graph is at most 2.

For a non-negative integer r, the r-orbit of a vertex v in a connected graph G of order n is the set of vertices at distance r from v. A sequence of positive integers with is called an irregular orbital dominating sequence of G if for every pair i, j of integers with and G contains distinct vertices such that We investigate graphs that possess and graphs do not possess an irregular orbital dominating sequence. It is shown that a non-trivial tree has an irregular orbital dominating sequence if and only if it is neither a star, a path of order 2, nor a path of order 6.

For an undirected graph $$G=(V,E)$$G=(V,E), a vertex $$\tau \in V$$źźV separates vertices u and v (where $$u,v\in V$$u,vźV, $$u\ne v$$uźv) if their distances to $$\tau $$ź are not equal. Given an integer parameter $$k \ge 1$$kź1, a set of vertices $$L\subseteq V$$L⊆V is a feasible solution, if for every pair of distinct vertices, u, v, there are at least k distinct vertices $$\tau _{1},\tau _{2},\ldots ,\tau _{k}\in L$$ź1,ź2,ź,źkźL, each separating u and v. Such a feasible solution is called a landmark set, and the k-metric dimension of a graph is the minimal cardinality of a landmark set for the parameter k. The case $$k=1$$k=1 is a classic problem, where in its weighted version, each vertex v has a non-negative cost, and the goal is to find a landmark set with minimal total cost. We generalize the problem for $$k \ge 2$$kź2, introducing two models, and we seek for solutions to both the weighted version and the unweighted version of this more general problem. In the model of all-pairs (AP), k separations are needed for every pair of distinct vertices of V, while in the non-landmarks model (NL), such separations are required only for pairs of distinct vertices in $$V \setminus L$$V\L. We study the weighted and unweighted versions for both models (AP and NL), for path graphs, complete graphs, complete bipartite graphs, and complete wheel graphs, for all values of $$k \ge 2$$kź2. We present algorithms for these cases, thus demonstrating the difference between the two new models, and the differences between the cases $$k=1$$k=1 and $$k \ge 2$$kź2.