Simple Undirected Graph Research Articles

On vertex-weighted realizations of acyclic and general graphs

Consider the following natural variation of the degree realization problem. Let G=(V,E) be a simple undirected graph of order n. Let f∈R≥0n be a vector of vertex requirements, and let w∈R≥0n be a vector of provided services at the vertices. Then w satisfies f on G if the constraints ∑j∈N(i)wj=fi are satisfied for all i∈V, where N(i) denotes the neighbourhood of vector i. Given a requirements vector f, the Vertex-Weighted Graph Realization problem asks for a suitable graph G and a vector w of provided services that satisfy f on G.In this paper, we consider two avenues. We initiate a study that focuses on weighted realizations where the graph is required to be of a specific class by providing a full characterization of realizable requirement vectors for paths and acyclic graphs. However, checking the respective criteria is shown to be NP-hard.In the second part, we advance the study in general graphs which was started in [2]. For the unsolved cases, the question of whether a vector f is realizable can be formulated as whether its largest requirement lies within certain intervals. We describe several new, realizable intervals and show the existence of an interval that cannot be realized. The complete classification for general graphs is an open problem.

Gorenstein and Cohen–Macaulay matching complexes

Let [Formula: see text] be a simple undirected graph. The family of all matchings of [Formula: see text] forms a simplicial complex called the matching complex of [Formula: see text]. Here, we give a classification of all graphs with a Gorenstein matching complex. Also we study when the matching complex of [Formula: see text] is Cohen–Macaulay and, in certain classes of graphs, we fully characterize those graphs which have a Cohen–Macaulay matching complex. In particular, we characterize when the matching complex of a graph with girth at least five or a complete graph is Cohen–Macaulay.

Finite groups whose prime graph on class sizes is a block square

Let G be a finite group, and let be the prime graph built on its set of conjugacy class sizes: this is the (simple undirected) graph whose vertices are the prime numbers dividing some conjugacy class size of G, and two distinct vertices p, q are adjacent if and only if pq divides some class size of G. In this paper, we characterize the structure of those groups G whose prime graph is a block square.

Genus two nilpotent graphs of finite commutative rings

Let [Formula: see text] be a finite commutative ring and [Formula: see text] is nilpotent for some [Formula: see text]. The nilpotent graph [Formula: see text] of [Formula: see text] is the simple undirected graph with vertex set [Formula: see text] in which two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] is nilpotent. In this paper, we observe a relationship between zerodivisor graphs, essential graphs and nilpotent graphs of [Formula: see text]. In continuation of genus characterizations in [T. Asir K. Mano and T. Tamizh Chelvam, Correction to: Classification of non-local rings with genus two zerodivisor graphs, Soft Comput. 25 (2021) 3355–3356; K. Selvakumar and M. Subajini, Finite commutative ring with genus two essential 10 graph, J. Algebra Appl. 16(2) (2018) 1850121], we classify all finite commutative rings (up to isomorphism) whose nilpotent graphs are of genus two.

Further results on the star degree of graphs

Let G be any simple undirected graph and let Q(G) be the signless Laplacian matrix of G. The polynomial ϕ(Q(G),x)=per(xI−Q(G)) is called the signless Laplacian permanental polynomial of G. The star degree of a graph G is the multiplicity of root 1 of ϕ(Q(G),x). Faria (1985) first considered the star degree of graphs. Based on Faria’s results, we further study the features of star degree of graphs, and give a formula to compute the star degree of a graph by a vertex partition of the graph. As applications, we derive the star degree set of n-vertex graphs, and we determine the graphs with extremal star degree. Furthermore, we show that some graphs with given star degree are determined by their signless Laplacian permanental spectra.

Relative g-noncommuting graph of finite groups

Let G be a finite group. For a fixed element g in G and a given subgroup H of G , the relative g -noncommuting graph of G is a simple undirected graph whose vertex set is G and two vertices x and y are adjacent if x ∈ H or y ∈ H and [ x , y ]≠ g , g −1 . We denote this graph by Γ H , G g . In this paper, we obtain computing formulae for degree of any vertex in Γ H , G g and characterize whether Γ H , G g is a tree, star graph, lollipop or a complete graph together with some properties of Γ H , G g involving isomorphism of graphs. We also present certain relations between the number of edges in Γ H , G g and certain generalized commuting probabilities of G which give some computing formulae for the number of edges in Γ H , G g . Finally, we conclude this paper by deriving some bounds for the number of edges in Γ H , G g .

Degree sum adjacency polynomial of standard graphs and graph operations

In this paper we explore the characteristic polynomials of degree sum adjacency matrix D S A ( G ) of a simple undirected graph G . We state a relation between the structure of a graph with the coefficients of its D S A polynomial. We obtain a generating function to find the number of walks of length k in a graph. Then, we obtain the degree sum adjacency polynomial for some standard graphs, derived graphs and for graph operations.

On the Number of k-Matchings in Graphs

In this paper, we present exact formulae for p(G, (3), p(G, (4) and p(G, (5) in terms of some degree-based invariants, where G is an undirected simple graph, a k-subset of edges in G without common vertices is called a k-matching and the number of such subsets is denoted by p(G, k). Significance of research work Molecular descriptors play a significant role in mathematical chemistry especially in QSPR/QSAR investigations. Among them, special place is reserved for so-called topological descriptors. Nowadays, there exists a legion of topological indices that found some applications in chemistry. In this paper, we obtain exact formulae for p(G, (3), p(G, (4) and p(G, (5) in terms of some degree-based invariants (topological indices).

Matching in Power Graphs of Finite Groups

The power graph P(G) of a finite group G is the undirected simple graph with vertex set G, where two elements are adjacent if one is a power of the other. In this paper, the matching numbers of power graphs of finite groups are investigated. We give upper and lower bounds, and conditions for the power graph of a group to possess a perfect matching. We give a formula for the matching number for any finite nilpotent group. In addition, using some elementary number theory, we show that the matching number of the enhanced power graph P_e(G) of G (in which two elements are adjacent if both are powers of a common element) is equal to that of the power graph of G.

The intersection graph of graded submodules of a graded module

Abstract In this article, we introduce and study the intersection graph of graded submodules of a graded module. Let M M be a left G G -graded R R -module. We define the intersection graph of G G -graded R R -submodules of M M , denoted by Γ ( G , R , M ) \Gamma \left(G,R,M) , to be a simple undirected graph whose set of vertices consists of all nontrivial G G -graded R R -submodules of M M , where two vertices are adjacent if their intersection is nonzero. We study properties of these graphs, such as connectivity, diameter, and girth. We also investigate the intersection graph of graded submodules for certain types of gradings such as faithful and strong gradings.

A polyhedral view to a generalization of multiple domination

Given an undirected simple graph G=(V,E) and integer values fv,v∈V, a node subset D⊆V is called an f-tuple dominating set if, for each node v∈V, its closed neighborhood intersects D in at least fv nodes. We study the polytope that is defined as the convex hull of the incidence vectors in RV of the f-tuple dominating sets in G. New families of valid inequalities are introduced and a complete formulation is given for the case of stars. A corollary of our results is a proof that the conjecture reported in Argiroffo (2013) on a complete formulation of the 2-tuple dominating set polytope of trees does not hold. Preliminary computational results are also reported.

Open-independent, open-locating-dominating sets: structural aspects of some classes of graphs

Let $G=(V(G),E(G))$ be a finite simple undirected graph with vertex set $V(G)$, edge set $E(G)$ and vertex subset $S\subseteq V(G)$. $S$ is termed \emph{open-dominating} if every vertex of $G$ has at least one neighbor in $S$, and \emph{open-independent, open-locating-dominating} (an $OLD_{oind}$-set for short) if no two vertices in $G$ have the same set of neighbors in $S$, and each vertex in $S$ is open-dominated exactly once by $S$. The problem of deciding whether or not $G$ has an $OLD_{oind}$-set has important applications that have been reported elsewhere. As the problem is known to be $\mathcal{NP}$-complete, it appears to be notoriously difficult as we show that its complexity remains the same even for just planar bipartite graphs of maximum degree five and girth six, and also for planar subcubic graphs of girth nine. Also, we present characterizations of both $P_4$-tidy graphs and the complementary prisms of cographs that have an $OLD_{oind}$-set.

Strongly Connected Orientation with Minimum Lexicographic Order of Indegrees

Given a simple undirected graph [Formula: see text], an orientation of [Formula: see text] is to assign every edge of [Formula: see text] a direction. Borradaile et al gave a greedy algorithm SC-Path-Reversal (in polynomial time) which finds a strongly connected orientation that minimizes the maximum indegree, and conjectured that SC-Path-Reversal is indeed optimal for the ”minimizing the lexicographic order” objective as well. In this note, we give a positive answer to the conjecture, which is that we show that the algorithm SC-PATH-REVERSAL finds a strongly connected orientation that minimizes the lexicographic order of indegrees.

On Reverse Laplacian Energy of a Graph

Let G be a simple undirected graph with n vertices and m edges. The Laplacian matrix L(G) of graph G is defined as L(G)=(l_ij), where (l_ij) is equal to -1 if v_i and v_j are adjacent, 1 if v_i and v_j are not adjacent and d(v_i) if i=j,where d(v_i) is the vertex degree of v_i. This paper defines the reverse Laplacian matrix, reverse Laplacian energy and find the same for some standard graphs. Further, we calculate upper for reverse Laplacian energy.

Reduced non-zero component union graph of free semi-modules

Let be a finitely generated free semimodule over a semiring with identity having the invariant basis number property. Let be a basis of and let Then the skeleton of a with respect to is defined as The reduced non-zero component union graph of with respect to is the simple undirected graph with vertex set where at least one of the ci ’s is zero and two distinct vertices are adjacent if and only if In this paper, we show that the graph is connected and find its domination number, clique number and chromatic number. In the case of finite semirings, we determine the order, size and degrees of vertices in Also, we give the characterization for planarity of

On the extended zero-divisor graph of strictly partial transformation semigroup

Given a commutative ring $R$, the zero-divisor graph of $R$ is an undirected simple graph with vertices the nonzero zero-divisors of $R$, and two distinct vertices $x$ and $y$ are adjacent if and only if $xy=0$. In [8], Redmond presented different versions of zero-divisor graphs of noncommutative rings. The main aim of this paper is to analyse these graphs for the semigroup $\mathcal{SP}_{n}$ of all strictly partial transformations on the set $X_{n}=\{1,2,\dots,n\}$.

Some properties of the topological spaces generated from the simple undirected graphs

Some properties of the topological spaces generated from the simple undirected graphs

Analysis of Eigenvalues for Molecular Structures

In this article, we study different molecular structures such as Polythiophene network, for and , Orthosilicate (Nesosilicate) , Pyrosilicates (Sorosilicates) , Chain silicates (Pyroxenes), and Cyclic silicates (Ring Silicates) for their cardinalities, chromatic numbers, graph variations, eigenvalues obtained from the adjacency matrices which are square matrices in order and their corresponding characteristics polynomials. We convert the general structures of these chemical networks in to mathematical graphical structures. We transform the molecular structures of these chemical networks which are mentioned above, into a simple and undirected planar graph and sketch them with various techniques of mathematics. The matrices obtained from these simple undirected graphs are symmetric. We also label the molecular structures by assigning different colors. Their graphs have also been studied for analysis. For a connected graph, the eigenvalue that shows its peak point (largest value) obtained from the adjacency matrix has multiplicity 1. Therefore, the gap between the largest and its smallest eigenvalues is interlinked with some form of “connectivity measurement of the structural graph”. We also note that the chemical structures, Orthosilicate (Nesosilicate) , Pyrosilicates (Sorosilicates) , Chain silicates (Pyroxenes) , and Cyclic silicates (Ring Silicates) generally have two same eigenvalues. Adjacency matrices have great importance in the field of computer science.

Crosscap two of class of graphs from commutative rings

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 structure of Cayley graphs of dihedral groups of Valencies 1, 2 and 3

Let G be a group and S be a subset of G such that e ∉ S and S−1 ⊆ S. Then Cay(G, S) is a simple undirected Cayley graph whose vertices are all elements of G and two vertices x and y are adjacent if and only if xy−1 ∈ S. The size of subset S is called the valency of Cay(G, S). In this paper, we determined the structure of all Cay(D2n, S), where D2n is a dihedral group of order 2n, n ≥ 3 and |S| = 1, 2 or 3.