Simple Undirected Graph Research Articles

Brouwer's Conjecture for the cartesian product of graphs

Let G be a simple, undirected graph with n vertices and m edges and Sk(G) the sum of its k largest Laplacian eigenvalues. Brouwer conjectured that Sk(G)≤m+(k+12) for any 1≤k≤n. We show that if G and H are simple undirected graphs satisfying Brouwer's Conjecture, then the cartesian product G□H also satisfies Brouwer's Conjecture.

Apriorics: Information and Graphs in the Description of the Fundamental Particles—A Mathematical Proof

In our earlier work, we suggested an axiomatic framework for deducing the fundamental entities which constitute the building block of the elementary particles in physics. The basic concept of this theory, named apriorics, is the ontological structure (OS)—an undirected simple graph satisfying specified conditions. The vertices of this graph represent the fundamental entities (FEs), its edges are binary compounds of the FEs (which are the fundamental bosons and fermions), and the structures constituting more than two connected vertices are composite particles. The objective of this paper is to focus the attention on several mathematical theorems and ideas associated with such graphs of order n, including their enumeration, showing what is the information content of apriorics.

CONDITIONS FOR GRAPHS ON <i>n</i> VERTICES WITH THE SUM OF DEGREES OF ANY TWO NONADJACENT VERTICES EQUAL TO<i> n-2</i> TO BE A HAMILTONIAN GRAPH

Let G be an undirected simple graph on \(n \geq 3\) vertices with the degree sum of any two nonadjacent vertices in G equal to \(n - 2\). We determine the condition for G to be a Hamiltonian graph.

Eulerian character degree graphs of solvable groups

Let G be a finite group, let Irr(G) be the set of all complex irreducible characters of G and let cd(G) be the set of all degrees of characters in Irr ( G ) . Let ρ ( G ) be the set of all primes that divide some degrees in cd ( G ) . The character degree graph Δ ( G ) of G is the simple undirected graph with vertex set ρ ( G ) and in which two distinct vertices p and q are adjacent if there exists a character degree r ∈ cd ( G ) such that r is divisible by the product pq. In this paper we obtain a necessary condition for the character degree graph Δ ( G ) of a solvable group G to be eulerian. We also prove that every n – 2 regular graph on n vertices where n is even and n ≥ 4 is the character degree graph of some solvable group. In the last part of the paper we give a bound for the number of Eulerian character degree graphs in terms of number of vertices.

G-Small Intersection Graph of a Module

Let be a commutative ring with identity, and be a left -module. The g-small intersection graph of non-trivial submodules of , indicated by , is a simple undirected graph whose vertices are in one-to-one correspondence with all non-trivial submodules of and two distinct vertices are adjacent if and only if the intersection of the corresponding submodules is a g-small submodule of . In this article, the interplay among the algebraic properties of , and the graph properties of are studied. Properties of such as connectedness, and completeness are considered. Besides, the girth and the diameter of are determined, as well as presenting a formula to compute the clique and domination numbers of . The graph is complete if, is a generalized hollow module or is a direct sum of two simple modules, is proved.

Universal adjacency spectrum of the cozero-divisor graph and its complement on a finite commutative ring with unity

For a finite simple undirected graph [Formula: see text], the universal adjacency matrix [Formula: see text] is a linear combination of the adjacency matrix [Formula: see text], the degree diagonal matrix [Formula: see text], the identity matrix [Formula: see text] and the all-ones matrix [Formula: see text], that is [Formula: see text], where [Formula: see text] and [Formula: see text]. The cozero-divisor graph [Formula: see text] of a finite commutative ring [Formula: see text] with unity is a simple undirected graph with the set of all nonzero nonunits of [Formula: see text] as vertices and two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] and [Formula: see text]. In this paper, we study structural properties of [Formula: see text] by defining an equivalence relation on its vertex set in terms of principal ideals of the ring [Formula: see text]. Then we obtain the universal adjacency eigenpairs of [Formula: see text] and its complement, and as a consequence one may obtain several spectra like the adjacency, Seidel, Laplacian, signless Laplacian, normalized Laplacian, generalized adjacency and convex linear combination of the adjacency and degree diagonal matrix of [Formula: see text] and [Formula: see text] in an unified way. Moreover, we get the universal adjacency eigenpairs of the cozero-divisor graph and its complement for a reduced ring and the ring of integers modulo [Formula: see text] in a simpler form.

Incidence Topological Spaces Generated from The Simple Undirected Graphs

In this paper, we investigate topologies produced by simple connected graphs. In particular, we associate a topology with G, called the incidence topology of G. A sub-base family to generate a incidence topology is implemented on the Vertices V set. Then we analyze some of the properties and discuss the impact topology of a few essential types of graphs. Our motivation in this section is to take a fundamental step towards the investigation of some of the characteristics of simple graphs by their corresponding incidence topology.

Prime labeling of graphs constructed from wheel graph

A prime labeling of a simple undirected graph G is to assign unique integer labels from the set {1,2,...,|V(G)|} to each vertex such that any two adjacent vertices in the graph have labels that are relatively prime. Studying prime labeling in graphs can help us understand the structure and properties of graphs and prime labeling has potential applications in cryptography and network security. In this paper we investigate when some graphs that are constructed from wheels are prime graphs.

Harmonic Mean Cordial Labeling of One Chord Cn V G

All the graphs considered in this article are simple and undirected. Let G = (V (G),E(G)) be a simple undirected graph. A function is called Harmonic Mean Cordial if the induced function defined by satisfies the condition and for any , where and denote the number of vertices and number of edges with label respectively and denotes the greatest integer less than or equals to . A graph is called a harmonic mean cordial graph if it admits harmonic mean cordial labeling. In this article, we have discussed the harmonic mean cordial labeling of One Chord .

Universal adjacency spectrum of the commuting and non-commuting graphs on AC-groups

For a simple undirected graph [Formula: see text] and [Formula: see text], [Formula: see text], the universal adjacency matrix [Formula: see text], where [Formula: see text] is the identity matrix, [Formula: see text] is the all-ones matrix, and [Formula: see text] and [Formula: see text] are the adjacency and degree diagonal matrices of [Formula: see text], respectively. For a finite non-abelian group [Formula: see text] and [Formula: see text], the commuting graph [Formula: see text] is a simple undirected graph with vertex set [Formula: see text] and the adjacency rule is commuting. The non-commuting graph of [Formula: see text] is the complement of [Formula: see text]. An AC-group is a non-abelian group such that the centralizer of every non-central element is abelian. In this paper, we obtain the universal adjacency eigenpairs of the commuting and non-commuting graphs for an arbitrary AC-group [Formula: see text] with [Formula: see text] or [Formula: see text]. Moreover, we determine the eigenpairs of [Formula: see text] when [Formula: see text] is a specific kind of group.

On the difference graph of power graphs of finite groups

The power graph of a finite group G is the simple undirected graph with vertex set G whose two vertices are adjacent if one is a power of the other. The enhanced power graph of a finite group G is the simple undirected graph whose vertex set is the group G whose two vertices a and b are adjacent if there exists c ∈ G such that both a and b are powers of c. In this paper, we investigate the difference graph Ɗ(G) of a finite group G, which is the difference of the enhanced power graph and the power graph of G with all isolated vertices removed. We first characterize an arbitrary finite group G such that Ɗ(G) is a chordal graph, star graph, dominatable, threshold graph, and split graph. From this, we conclude that the latter four graph classes are equal for Ɗ(G). By applying these results, we classify the nilpotent groups G such that Ɗ(G) belong to the aforementioned five graph classes. This shows that all these graph classes are equal for Ɗ(G) when G is nilpotent. Then, we characterize the nilpotent groups whose difference graphs are cograph, bipartite, Eulerian, planar, and outerplanar. Finally, we consider the difference graph of non-nilpotent groups and determine the values of n such that the difference graphs of the symmetric group Sn and alternating group An are cograph, chordal, split, and threshold.

On maximal block topological space

In this paper we create a new topological structure induced by connected simple undirected graphs called maximal block topological space and study some properties of this new type of topology. Also, define some concepts in maximal block topological space like (derived subgraph, closure subgraph and interior subgraph). Some results and properties of vertices and subgraphs in G due to maximal block topological space are proved and discussed. Moreover, showed that a maximal block topological space is T0-space and T1/2-space if and only if G is acyclic graph. Finally, irreducibility and topologically independent of maximal block topological space are introduced.

Iterated Clique Reductions in Vertex Weighted Coloring for Large Sparse Graphs.

The Minimum Vertex Weighted Coloring (MinVWC) problem is an important generalization of the classic Minimum Vertex Coloring (MinVC) problem which is NP-hard. Given a simple undirected graph G=(V,E), the MinVC problem is to find a coloring s.t. any pair of adjacent vertices are assigned different colors and the number of colors used is minimized. The MinVWC problem associates each vertex with a positive weight and defines the weight of a color to be the weight of its heaviest vertices, then the goal is the find a coloring that minimizes the sum of weights over all colors. Among various approaches, reduction is an effective one. It tries to obtain a subgraph whose optimal solutions can conveniently be extended into optimal ones for the whole graph, without costly branching. In this paper, we propose a reduction algorithm based on maximal clique enumeration. More specifically our algorithm utilizes a certain proportion of maximal cliques and obtains lower bounds in order to perform reductions. It alternates between clique sampling and graph reductions and consists of three successive procedures: promising clique reductions, better bound reductions and post reductions. Experimental results show that our algorithm returns considerably smaller subgraphs for numerous large benchmark graphs, compared to the most recent method named RedLS. Also, we evaluate individual impacts and some practical properties of our algorithm. Furthermore, we have a theorem which indicates that the reduction effects of our algorithm are equivalent to that of a counterpart which enumerates all maximal cliques in the whole graph if the run time is sufficiently long.

The change of Seidel energy of 5-partite Turán graph due to edge deletion

Let G be a simple undirected graph. The Seidel energy of G, denoted by Es(G), is the sum of absolute values of all Seidel eigenvalues of G. A complete r-partite graph with n vertices is called r-partite Turán graph, denoted by T(n,r), if the number of vertices of each part is either ⌊nr⌋ or ⌈nr⌉. In 2021, Tian et al. proposed a problem that for any edge e of Turán graph T(n,r) with r≥4, does there exist a sufficiently large integer n0 such that Es(T(n,r))<Es(T(n,r)−e) whenever n≥n0. In this paper, we prove that the Seidel energy of T(n,5) of order n≥5 is always increased when an edge is deleted.

Graceful Labeling of Prime Index Graph of Group Zp × Zpn

The prime index graph π(G) of a finite group G is a special type of undirected simple graph whose vertex set is set of subgroups of G, in which two distinct vertices are adjacent if one has prime index in the other. Let p and q be distinct primes. In this paper, we establish that prime index graph of a finite cyclic p-group Zpn, a finite abelian group Zpn × Zq and a finite abelian p-group Zp × Zpn always have graceful labeling without any condition on n using the concept of path graph or p-layer ladder graph of size n + 1.

Characterization of rings with genus two prime ideal sum graphs

Let [Formula: see text] be a commutative ring with unity. The prime ideal sum graph of the ring [Formula: see text] is a simple undirected graph whose vertex set is the set of nonzero proper 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] is a prime ideal of [Formula: see text]. In this paper, we characterize all the finite non-local commutative rings whose prime ideal sum graph is of genus [Formula: see text].

THE NUMBER OF CC-DOMINATING SETS OF SOME GRAPHS

The aim of this paper is to study about the number of cc-dominating sets and to introduce the concept of cc-domination polynomial for simple finite undirected graphs. For a graph G on n vertices possessing dc (G,i) cc-dominating sets of cardinality i, the cc-domination polynomial is defined as Dc [G;x]= n i=γcc (G) dc (G,i) xi, whereγ cc(G) is the cc-domination number of G. We obtain some properties of Dc[G;x] and compute the same for some special graphs. Moreover, the concept of cc-domination entropy is also introduced and studied.

On the Inclusion Ideal Graph of Semigroups

The inclusion ideal graph [Formula: see text] of a semigroup [Formula: see text] is an undirected simple graph whose vertices are all the nontrivial left ideals of [Formula: see text] and two distinct left ideals [Formula: see text], [Formula: see text] are adjacent if and only if either [Formula: see text] or [Formula: see text]. The purpose of this paper is to study algebraic properties of the semigroup [Formula: see text] as well as graph theoretic properties of [Formula: see text]. We investigate the connectedness of [Formula: see text] and show that the diameter of [Formula: see text] is at most 3 if it is connected. We also obtain a necessary and sufficient condition of [Formula: see text] such that the clique number of [Formula: see text] is the number of minimal left ideals of [Formula: see text]. Further, various graph invariants of [Formula: see text], viz. perfectness, planarity, girth, etc., are discussed. For a completely simple semigroup [Formula: see text], we investigate properties of [Formula: see text] including its independence number and matching number. Finally, we obtain the automorphism group of [Formula: see text].

A study of upper ideal relation graphs of rings

Let R be a ring with unity. The upper ideal relation graph of the ring R is the simple undirected graph whose vertex set is the set of all non-unit elements of R and two distinct vertices x, y are adjacent if and only if there exists a non-unit element such that the ideals (x) and (y) contained in the ideal (z). In this article, we obtain the girth, minimum degree and the independence number of . We obtain a necessary and sufficient condition on R, in terms of the cardinality of their principal ideals, such that the graph is planar and outerplanar, respectively. For a non-local commutative ring , where Ri is a local ring with maximal ideal and , we prove that the graph is perfect if and only if and each is a principal ideal. We also discuss all the finite rings R such that the graph is Eulerian. Moreover, we obtain the metric dimension and strong metric dimension of , when R is a reduced ring. Finally, we determine the vertex connectivity, automorphism group, Laplacian and the normalized Laplacian spectrum of . We classify all the values of n for which the graph is Hamiltonian.

Adjacency matrices over a finite prime field and their direct sum decompositions

In this paper, we discuss the adjacency matrices of finite undirected simple graphs over a finite prime field Fp. We apply symmetric (row and column) elementary transformations to the adjacency matrix over Fp in order to get a direct sum decomposition by other adjacency matrices. In this paper, we give a complete description of the direct sum decomposition of the adjacency matrix of any graph over Fp for any odd prime p. Our key tool is quadratic residues of Fp.