Simple Undirected Graph Research Articles

Certain properties of the enhanced power graph associated with a finite group

The enhanced power graph of a finite group $$G$$ , denoted by $$P_E(G)$$ , is a simple undirected graph whose vertex set is G and two distinct vertices x, y are adjacent if $$x, y \in \langle z \rangle$$ for some $$z \in G$$ . In this article, we determine all finite groups such that the minimum degree and the vertex connectivity of $$P_E(G)$$ are equal. Also, we classify all groups whose (proper) enhanced power graphs are strongly regular. Further, the vertex connectivity of the enhanced power graphs associated to some nilpotent groups is obtained. Finally, we obtain the upper and lower bounds of the Wiener index of $$P_E(G)$$ , where G is a nilpotent group. The finite nilpotent groups attaining these bounds are also characterized.

Commuting graph of CA−groups

A group G is called a CA−group, if all the element centralizers of G are abelian and the commuting graph of G with respect to a subset A of G, denoted by Γ(G, A), is a simple undirected graph with vertex set A and two distinct vertices a and b are adjacent if and only if ab = ba. The aim of this paper is to generalize results of a recently published paper of F. Ali, M. Salman and S. Huang [On the commuting graph of dihedral group, Comm. Algebra 44 (6) (2016) 2389—2401] to the case that G is an CA−group.

More on Result Involution Graphs

The result involution graph of a finite group , denoted by is an undirected simple graph whose vertex set is the whole group and two distinct vertices are adjacent if their product is an involution element. In this paper, result involution graphs for all Mathieu groups and connectivity in the graph are studied. The diameter, radius and girth of this graph are also studied. Furthermore, several other graph properties are obtained.

Hop Differentiating Hop Dominating Sets in Graphs

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.

On the Laplacian spread of digraphs

In this article, we extend the notion of the Laplacian spread to simple directed graphs (digraphs) using the restricted numerical range. First, we provide Laplacian spread values for several families of digraphs. Then, we prove sharp upper bounds on the Laplacian spread for all polygonal and balanced digraphs. In particular, we show that the validity of the Laplacian spread bound for balanced digraphs is equivalent to the Laplacian spread conjecture for simple undirected graphs, which was conjectured in 2011 and proven in 2021. Moreover, we prove an equivalent statement for weighted balanced digraphs with weights between 0 and 1. Finally, we state several open conjectures that are motivated by empirical data.

Non-solvable groups whose character degree graph has a cut-vertex. II

Let G be a finite group, and let $$\textrm{cd}(G)$$ denote the set of degrees of the irreducible complex characters of G. Define then the character degree graph $$\Delta (G)$$ as the (simple undirected) graph whose vertices are the prime divisors of the numbers in $$\textrm{cd}(G)$$ , and two distinct vertices p, q are adjacent if and only if pq divides some number in $$\textrm{cd}(G)$$ . This paper continues the work, started in [8], toward the classification of the finite non-solvable groups whose degree graph possesses a cut-vertex, i.e. a vertex whose removal increases the number of connected components of the graph. While, in [8], groups with no composition factors isomorphic to $$\textrm{PSL}_{2}(t^a)$$ (for any prime power $$t^a\ge 4$$ ) were treated, here we consider the complementary situation in the case when $$t$$ is odd and $$t^a> 5$$ . The proof of this classification will be then completed in the third and last paper of this series [7] that deals with the case $$t=2$$ .

Cayley subspace sum graph of vector spaces

Let $\mathbb{V}$ be a finite dimensional vector space over the field $\mathbb{F}$. Let $S(\mathbb{V})$ be the set of all subspaces of $\mathbb{V}$ and $\mathbb{A}\subseteq S^*(\mathbb{V})=S(\mathbb{V})\backslash\{0\}.$ In this paper, we define the Cayley subspace sum graph of $\mathbb{V},$ denoted by Cay$(S^*(\mathbb{V}),\mathbb{A}), $ as the simple undirected graph with vertex set $S^*(\mathbb{V})$ and two distinct vertices $X$ and $Y$ are adjacent if $X+Z=Y$ or $Y+Z=X$ for some $Z\in \mathbb{A}$. Having defined the Cayley subspace sum graph, we study about the connectedness, diameter and girth of several classes of Cayley subspace sum graphs Cay$(S^*(\mathbb{V}), \mathbb{A})$ for a finite dimensional vector space $\mathbb{V}$ and $\mathbb{A}\subseteq S^*(\mathbb{V})=S(\mathbb{V})\backslash\{0\}.$

A study on determination of some graphs by Laplacian and signless Laplacian permanental polynomials

The permanent of an n × n matrix is defined as where the sum is taken over all permutations σ of The permanental polynomial of M, denoted by is where In is the identity matrix of order n. Let G be a simple undirected graph on n vertices and its Laplacian and signless Laplacian matrices be L(G) and Q(G) respectively. The permanental polynomials and are called the Laplacian permanental polynomial and signless Laplacian permanental polynomial of G respectively. A graph G is said to be determined by its (signless) Laplacian permanental polynomial if all the graphs having the same (signless) Laplacian permanental polynomial with G are isomorphic to G. A graph G is said to be combinedly determined by its Laplacian and signless Laplacian permanental polynomials if all the graphs having (i) the same Laplacian permanental polynomial as and (ii) the same signless Laplacian permanental polynomial as are isomorphic to G. In this article we investigate the determination of some graphs, namely, star, wheel, friendship graphs and a particular kind of caterpillar graph (whose all r non-pendant vertices have the same degree n) by their Laplacian and signless Laplacian permanental polynomials. We show that a kind of caterpillar graphs (for ), wheel graph (up to 7 vertices) and friendship graph (up to 7 vertices) are determined by their (signless) Laplacian permanental polynomials. Further we prove that all friendship graphs and wheel graphs are combinedly determined by their Laplacian and signless Laplacian permanental polynomials.

Construction of the <img src=image/13492345_01.gif> Graph of Mathieu Group <img src=image/13492345_02.gif>

Suppose that <img src=image/13492345_03.gif> is a group and <img src=image/13492345_04.gif> is a subset of <img src=image/13492345_03.gif>. Then, the <img src=image/13492345_05.gif> graph of a group <img src=image/13492345_03.gif>, denoted by <img src=image/13492345_06.gif>, is the simple undirected graph in which two distinct vertices <img src=image/13492345_07.gif> are connected to each other by an edge if and only if both vertices satisfy <img src=image/13492345_08.gif>. The main contribution of this paper is to construct the <img src=image/13492345_05.gif> graph using the elements of Mathieu group, <img src=image/13492345_09.gif>. Additionally, the connectivity of <img src=image/13492345_06.gif> has been proven as a connected graph. Finally, an open problem is highlighted in addressing future research.

The topological space generated from the graphs by use of some relation

In this article, we study the topology and its properties generated from a simple undirected graph with no isolated vertices. We use a simple undirected graph to generate the topology, which connects event edges with the neighboring relationship between vertices, edges, and studies some properties of the graph.

Studying the result involution graphs for HS and McL Leech lattice groups

Let G be a finite group, the result is the involution graph of G, which is an undirected simple graph denoted by the group G as the vertex set and x, y ∈ G adjacent if xy and (xy)2 = 1. In this article, we investigate certain properties of G, the Leech lattice groups HS and McL. The study involves calculating the diameter, the radius, and the girth of ΓGRI.

A STUDY OF AN UNDIRECTED GRAPH ON A FINITE SUBSET OF NATURAL NUMBERS

Let $G_{n}=(V,E)$ be an undirected simple graph, whose vertex set comprises of the natural numbers which are less than $n$ but not relatively prime to $n$ and two distinct vertices $u,v \in V$ are adjacent if and only if $\gcd(u,v)&gt;1$. Connectedness, completeness, minimum degree, maximum degree, independence number, domination number and Eulerian property of the graph $G_n$ are studied in this paper.

On the Essential Element Graph of a Lattice

Let $\mathcal{L}$ be a bounded lattice. The essential element graph of $\mathcal{L}$ is a simple undirected graph $\varepsilon_{\mathcal{L}}$ such that the elements $x,y$ of $\mathcal{L}$ form an edge in $\varepsilon_{\mathcal{L}}$, whenever $x \vee y $ is an essential element of $\mathcal{L}$. In this paper, we study properties of the essential elements of lattices and essential element graphs. We study the lattices whose zero-divisor graphs and incomparability graphs are isomorphic to its essential element graphs. Moreover, the line essential element graphs are investigated.

On the weakly nilpotent graph of a commutative semiring

Let S be a commutative semiring with unity. In this paper, we introduce the weakly nilpotent graph of a commutative semiring. The weakly nilpotent graph of S, denoted by Γw(S) is defined as an undirected simple graph whose vertices are S and two distinct vertices x and y are adjacent if and only if xy 2 N(S), where S= Sn f0g and N(S) is the set of all non-zero nilpotent elements of S. In this paper, we determine the diameter of weakly nilpotent graph of an Artinian semiring. We prove that if w(S) is a forest, then Γw(S) is a union of a star and some isolated vertices. We study the clique number, the chromatic number and the independence number of Γw(S). Among other results, we show that for an Artinian semiring S, Γw(S) is not a disjoint union of cycles or a unicyclic graph. For Artinian semirings, we determine diam(Γw(S)). Finally, we characterize all commutative semirings S for which Γw(S) is a cycle, where w(S) is the complement of the weakly nilpotent graph of S. Finally, we characterize all commutative semirings S for which Γw(S) is a cycle.

Graph-based confidence verification for BPSK signal analysis under low SNRs

Confidence verification for intercepted signal analysis has important military applications where a priori knowledge of the signal or transmission channel is insufficient or absent. However, existing methods do not perform well under low signal-to-noise ratios (SNRs). In this study, we propose a graph-based verification algorithm for binary phase shift keying (BPSK) signal analysis. The correlation spectrum between the original observed signal and reference signal can be divided into blocks of equal size and the summation of each block (called the block summation (BS) spectrum) obtained. Then, the BS spectrum samples are transformed into a specific simple undirected graph using the signal-to-graph convertor. Accordingly, the verification issue can be simplified to determining whether the graph is fully-connected. The majorization relationships between the probability density functions of the BS and the original correlation spectra are evaluated. This is used to compare the connectivity of the graphs generated from those two types of samples, as well as their probabilities of being complete graphs. Simulation results confirm the superior performance of the proposed algorithm over previous confidence verification algorithms for low SNRs and in the presence of transmission impairments, as well as a lower computational complexity when compared with the conventional graph-based algorithm.

On the spectra of non-commuting graphs of certain class of groups

Let [Formula: see text] be the center of a finite non-abelian group [Formula: see text] The non-commuting graph of [Formula: see text] is a simple undirected graph with vertex set [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] A group [Formula: see text] is called a CA-group if the centralizer [Formula: see text] for every non central element [Formula: see text] is abelian. In this paper, we investigate the distance, distance (signless) Laplacian spectra of non-commuting graphs of some CA-groups, and obtain some conditions on them so that the corresponding non-commuting graph is distance, distance (signless) Laplacian integral.

GENERALIZED NON-BRAID GRAPHS OF RINGS

In this paper, we introduce the definition of generalized non-braid graph of a given ring. Let $R$ be a ring and let $k$ be a natural number. By generalized braider of $R$ we mean the set $B^k(R):=\{x \in R~|~\forall y \in R,~ (xyx)^k = (yxy)^k\}$. The generalized non-braid graph of $R$, denoted by $G_k(\Upsilon_R)$, is a simple undirected graph with vertex set $R\backslash B^k(R)$ and two distinct vertices $x$ and $y$ are adjacent if and only if $(xyx)^k \neq (yxy)^k$. In particular, we investigate some properties of generalized non-braid graph $G_k(\Upsilon_{\mathbb{Z}_n})$ of the ring $\mathbb{Z}_n$.

Reconstruction of helm graph and web graph

The graph reconstruction conjecture asserts that every simple undirected graph with number of vertices (where <em>n</em> ≥ 3) is uniquely determined up to isomorphism, by its collection of unlabeled form of vertex deleted sub graphs. If a vertex deleted sub graph of <em>G</em> is given in unlabeled form, then it is called a “card” of <em>G</em>. The collection of all cards of <em>G</em> is called the “deck” of <em>G</em>, and is denoted by <em>D</em>(<em>G</em>) . If we can determine the original graph <em>G</em> from the deck of the graph <em>G</em>, we can say that it is reconstructible. In this paper we propose a new method to reconstruct the helm graph (<em>H_n</em>) and the web graph (<em>W_n</em>), by using the degree sequence of its collection of vertex deleted sub graphs. Helm graph is obtained by adjoining a pendant edge to each node of the cycle of the -wheel graph. Web graph is obtained by joining pendant vertices to the vertices in the outer cycle of the helm graph to form a cycle and again adding pendant vertices to the new cycle. For both graphs we found decks and obtained the degree sequence of each cards. Finally, we reconstructed the original graph using the degree sequences.

On the genus of reduced cozero-divisor graph of commutative rings

Let R be a commutative ring with identity and let (x) be the principal ideal generated by \(x\in R.\) Let \(\varOmega (R)^*\) be the set of all nontrivial principal ideals of R. The reduced cozero-divisor graph \(\varGamma _r(R)\) of R is an undirected simple graph with \(\varOmega (R)^*\) as the vertex set and two distinct vertices (x) and (y) in \(\varOmega (R)^*\) are adjacent if and only if \((x)\nsubseteq (y)\) and \((y)\nsubseteq (x)\). In this paper, we characterize all classes of commutative Artinian non-local rings for which the reduced cozero-divisor graph has genus at most one.

On Some Properties of Non-traceable Cubic Bridge Graph

Graphs considered in this paper are simple finite undirected graphs without loops or multiple edges. A simple graphs where each vertex has degree 3 is called cubic graphs. A cubic graphs, that is, 1-connected or cubic bridge graph is traceable if its contains Hamiltonian path otherwise, non-traceable. In this paper, we introduce a new family of cubic graphs called Non-Traceable Cubic Bridge Graph (NTCBG) that satisfy the conjecture of Zoeram and Yaqubi (2017). In addition, we defined two important connected component of NTCBG that is, central fragment that give assurance for a graph to be non-traceable and its branches. Some properties of NTCBG such as chromatic number and clique number were also provided.