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Let S be a groupoid (magma) with zero 0, and let R = ⊕ s ∈ S R s be a contracted S-graded ring, that is, an S-graded ring with R 0 = 0. By G (H R) we denote the undirected power graph of a multiplicative subsemigroup H R = ∪ s ∈ S R s of R, and by G * (H R) a graph obtained from G (H R) by removing 0 and its incident edges. If Re is a nonzero ring component of R, then G * (R e) denotes a subgraph of G * (H R) , induced by R e *. In this paper we address a problem raised in [Abawajy, J., Kelarev, A., Chowdhury, M.: Power Graphs: A Survey. Electron. J. Graph Theory Appl. 1(2), 125–147 (2013)]. Namely, let S be torsion-free, that is, s n = t n implies s = t for all s, t ∈ S , and all positive integers n, and let S be 0-cancellative, that is, for all s, t, u ∈ S , su = tu ≠ 0 implies s = t , and us = ut ≠ 0 implies s = t. Also, let R be semisimple Artinian. We prove that if G * (R e) is connected for every nonzero ring component Re of R, then the connected components of G * (H R) are precisely the graphs G * (R e). [ABSTRACT FROM AUTHOR]

In this paper we introduce Cayley digraphs associated to finitely generated polygroups, where the vertices correspond to finite products of the generators of polygroups and the edges to multiplication by vertices and generators. We investigate some properties of the Cayley digraphs, emphasizing connectivity and existence of cycles for each vertex of the Cayley digraphs. Particularly, we identify Cayley digraphs on polygroups derived from conjugate classes of dihedral groups. Moreover, we examine some fundamental illustrative examples of Cayley digraphs through the class of gmg-polygroups. [ABSTRACT FROM AUTHOR]

A simple graph G on n vertices is partition-good if and only if for all pairs (a, b) of positive integers such that a+b = n, V (G) can be partitioned into sets A,B satisfying: |A| = a, |B| = b, and G[A],G[B] are connected. G is partition-wonderful if and only if either (i) n = 1, or (ii) n > 1, G is connected, and of all pairs (a, b) of positive integers such that a + b = n, V (G) can be partitioned into sets A,B satisfying: |A| = a, |B| = b, and G[A],G[B] are partition-wonderful. We characterize the partition-good and the partition-wonderful among the complete multipartite graphs, and, along the way, prove some elementary results about these graph properties. [ABSTRACT FROM AUTHOR]

Molecular structures are characterised by the Hosoya polynomial and Wiener index, ideas from mathematical chemistry and graph theory. The graph representation of a chemical compound that has atoms as vertices and chemical bonds as edges is called a molecular graph, and the Hosoya polynomial is a polynomial related to this graph. As a graph attribute that remains unchanged under graph isomorphism, the Hosoya polynomial is known as a graph invariant. It offers details regarding the quantity of distinct non-empty subgraphs within a specified graph. A topological metric called the Wiener index is employed to measure the branching complexity and size of a molecular graph. For every pair of vertices in a molecular network, the Wiener index is the total of those distances. In this paper, discussed the Hosoya polynomial, Wiener index and Hyper-Wiener index of the Abid-Waheed graphs (AW)a8  and (AW)a10. This graph is similar to Jahangir’s graph. Further, we have extended the research work on the applications of the described graphs. [ABSTRACT FROM AUTHOR]

Let Γ = (V, E) be a non-trivial planar connected graph with vertex set V and edge set E. A set of ordered vertices R from V(Γ) is said to be a resolving set for Γ if each vertex of Γ is uniquely determined by its vector of distances to the vertices of R. The number of vertices in a smallest resolving set is called the metric dimension of Γ. In this article, we study the metric dimension for two families of planar graphs, each of which is shown to have an independent minimum resolving set with cardinality three. [ABSTRACT FROM AUTHOR]

Graph theory have strong connections with other fields of mathematics. In this article we introduce two new concepts in graph theory by collaborating with number theory and real analysis. In the second section, a new graph product called the ARMS-product of graphs is introduced, in which the degree of each vertex and the greatest common divisor of these degrees determines the adjacency of the graph. ARMS-product of certain classes of graphs including regular graphs, paths, cycles, stars are determined. In section 3 we associate the concept of limit in real analysis with graph theory by defining a new graph called limit graph of a given graph. Limit graphs of certain standard graphs as well as graphs obtained through various graph theoretic operations are derived with suitable illustrations. Further we determine some graphs whose limit graphs are unique as well as not unique. [ABSTRACT FROM AUTHOR]

Let R be a ring with nonzero identity. The coidempotent graph of R , denoted it by G I d − (R) , has its set of vertices equal to the set of all elements of R ; distinct vertices x , y are adjacent in G I d − (R) if and only if x − y or y − x is an idempotent of R. In this paper, we study some basic properties of G I d − (R) and find some results about the ring-theoretic properties of R and graph-theoretic properties of G I d − (R). [ABSTRACT FROM AUTHOR]

In this paper, we define the notion of (F - H)G-contraction and utilize the same to obtain fixed point results in the setting of metric spaces equipped with a graph. Our contraction generalizes many known contractions in literature. As an application, we have proved the existence and uniqueness of solution of Volterra type integral equation. Finally, some suitable examples are given to validate our claim. [ABSTRACT FROM AUTHOR]

The proposed method is based on calculations of the statistics of the nearest neighbor graph (NNG) structures, which are presented as a benchmark of the probabilities of the distribution of graphs by the number of disconnected fragments. The deviation of the actually observed occurrence of connectivity from the calculated one will allow us to determine the probability that this sample can be considered a set of statistically independent variables. The statements about the independence of the NNG statistics from the distribution of distances and from the triangle inequality are proved, which allows the numerical modeling of such structures. Estimates of the accuracy of the calculated statistics for graphs and their comparison with estimates obtained by modeling random coordinates of points in d-dimensional space are carried out. It is shown that the model of the NNGs without taking into account the dimension of the space leads to fairly accurate estimates of the statistics of graph structures in spaces of dimensionality higher than five. For spaces of smaller dimensionality, the benchmark can be obtained by directly calculating the distances between points with random coordinates in a unit cube. The proposed method is applied to the problem of analyzing the level of unsteadiness of the earthquake catalog in the Kuril–Kamchatka region. The lengths of samples of time intervals between neighboring events are analyzed. It is shown that the analyzed system as a whole is interconnected with a probability of 0.91, and this dependence is fundamentally different from the lag correlation between the sample elements. [ABSTRACT FROM AUTHOR]

Let A be a nonzero commutative ring with identity. Let n be a positive integer. Let  R = A × ⋯ × A (n times). The total dot product graph of R denoted by TD (R) is an undirected graph with vertex set which equals R ∖ { (0 , ... , 0) } and distinct vertices x and y are adjacent if and only if x ⋅ y = 0 , where x ⋅ y is the normal dot product of x and y. Let Z (R) denote the set of all zero-divisors of R and let us denote Z (R) ∖ { (0 , ... , 0) } by Z (R) * . The zero-divisor dot product graph of R , denoted by ZD (R) is the subgraph of TD (R) induced by Z (R) * . The graph TD (R) (respectively, ZD (R)) was introduced and investigated by Badawi [Commun. Algebra 43(1) (2015) 43–50]. In this paper, we characterize R such that (TD (R)) c (respectively, (ZD (R)) c ) is connected and determine the diameter and the radius of (TD (R)) c (respectively, (ZD (R)) c ) whenever it is connected. Moreover, if (TD (R)) c (respectively, (ZD (R)) c ) is connected, then we characterize R such that (TD (R)) c (respectively, (ZD (R)) c ) admits a cut vertex. [ABSTRACT FROM AUTHOR]

Let \( R \) be a finite commutative ring with unity. We have introduced a divisor graph of the ring, denoted by \( D[R] \). It is an undirected simple graph with a vertex set \( V = R - \{0, 1\} \), and two distinct vertices \( u \) and \( v \) in \( V \) are adjacent if and only if there exists \( w \in R \) such that \( v = uw \) or \( u = vw \). Clearly, \( 0 \) and unit elements are adjacent to all elements of the ring. Thus, we adopt the idea of Anderson and Livingston and remove the zero and unit vertices from the vertex set. A new graph is called the modified divisor graph, noted by symbol \( D_0[R] \).In this study, we have focused on the structural properties of \( D_0[\mathbb{Z}_n] \). Moreover, we have determined the diameter, girth, clique number, vertex connectivity, and edge connectivity of \( D_0[\mathbb{Z}_n] \) for every natural number \( n \). The purpose of studying this graph is to find relationships between the ring theoretic properties of \( R \) and the graph theoretic properties of \( D_0[R] \).

In this work, a new class of general contractive mappings, with the name Jaggi-Suzuki-type hybrid (K-α-ϕ)-contractive mapping is discussed in metric space equipped with a graph and new criteria for which the mapping is a Picard operator are studied. The superiority of this type of contractive mapping lies in the fact that its contractive inequality can be fixed in different ways, depending on the specified constants. Substantial illustrations are furnished to validate the axioms of our obtained ideas and to show their difference from the existing concepts. Supplementarily, some corollaries which collapse our obtained notion to recently propounded results in the literature are brought out and analysed.

In this paper, some properties of locally antisymmetrically connected spaces which are the localized version of the antisymmetrically connected $T_0$-quasi-metric spaces constructed as the natural counterparts of connected complementary graphs, are presented in terms of asymmetric norms. According to that, we investigated some different aspects and examples of local antisymmetric connectedness in the framework of asymmetrically normed real vector spaces. Specifically, it is proved that the structures of antisymmetric connectedness and local antisymmetric connectedness coincide for the $T_0$-quasi-metrics induced by the asymmetric norms which associate the theory of quasi-metrics with functional analysis.

In this paper we introduce Cayley digraphs associated to finitely generated polygroups, where the vertices correspond to finite products of the generators of polygroups and the edges to multiplication by vertices and generators. We investigate some properties of the Cayley digraphs, emphasizing connectivity and existence of cycles for each vertex of the Cayley digraphs. Particularly, we identify Cayley digraphs on polygroups derived from conjugate classes of dihedral groups. Moreover, we examine some fundamental illustrative examples of Cayley digraphs through the class of gmg-polygroups.

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.

For a connected n -vertex graph G and a positive integer k , let ι k (G) denote the size of a smallest set D of vertices of G such that the graph obtained from G by deleting the closed neighbourhood of D contains no connected graph that has at least k edges. By a result of Caro and Hansberg, ι 1 (G) ≤ n / 3 if n ≠ 2 and G is not a 5-cycle. Let r be the number of vertices of G that have only one neighbour. We show that ι 2 (G) ≤ (4 n − r) / 14 if G is not a copy of one of six graphs. We also show that ι 3 (G) ≤ n / 4 if G is neither a triangle nor a 7-cycle. The bounds are sharp. The two new results imply recent results on isolation of graphs. The bound on ι 3 (G) strengthens the author's solution to a problem of Caro and Hansberg on isolation of cycles. [ABSTRACT FROM AUTHOR]

For a graph G and a family of graphs $\mathcal {F}$ , the Turán number ${\mathrm {ex}}(G,\mathcal {F})$ is the maximum number of edges an $\mathcal {F}$ -free subgraph of G can have. We prove that ${\mathrm {ex}}(G,\mathcal {F})\ge {\mathrm {ex}}(K_r, \mathcal {F})$ if the chromatic number of G is r and $\mathcal {F}$ is a family of connected graphs. This result answers a question raised by Briggs and Cox ['Inverting the Turán problem', Discrete Math. 342 (7) (2019), 1865–1884] about the inverse Turán number for all connected graphs. [ABSTRACT FROM AUTHOR]

The structures of symmetric connectedness and dually, antisymmetric connectedness were described and studied before, especially in terms of graph theory as the corresponding counterparts of the connectedness of a graph and the connectedness of its complementary graph. By taking into consideration the deficiencies of topological density in the context of symmetric and antisymmetric connectedness, two special kinds of density in the theory of non-metric T0-quasi-metrics were introduced in the previous studies under the names symmetric density and antisymmetric density. In this paper, some crucial and useful properties of these two types of density are investigated with the help of the major results and (counter)examples peculiar to the asymmetric environment. Besides these, many further observations about the structures of symmetric and antisymmetric-density are dealt with, especially in the sense of their combinations such as products and unions through various theorems in the context of T0-quasi-metrics. Also, we examine the question of under what kind of quasi-metric mapping these structures will be preserved. [ABSTRACT FROM AUTHOR]

Let RG be the group ring of the group G over a ring R and let Z*(RG) be the collection of all non-zero zero divisors in a finite group ring RG. And, for x ∈ Z*(RG), ann(x) = {r ∈ RG/rx = 0}. The annihilator graph of RG denoted as AG(RG), and is defined as the graph whose vertex set is the elements of non-zero zero divisors in RG and two distinct vertices vx and vy are adjacent if and only if ann(x) ∩ ann(y) ̸= {0}. In this paper we try to characterize the properties of annihilator graphs in group rings. [ABSTRACT FROM AUTHOR]

In this paper, some properties of locally antisymmetrically connected spaces which are the localized version of the antisymmetrically connected T0-quasi-metric spaces constructed as the natural counterparts of connected complementary graphs, are presented in terms of asymmetric norms. According to that, we investigated some different aspects and examples of local antisymmetric connectedness in the framework of asymmetrically normed real vector spaces. Specifically, it is proved that the structures of antisymmetric connectedness and local antisymmetric connectedness coincide for the T0-quasi-metrics induced by the asymmetric norms which associate the theory of quasi-metrics with functional analysis. [ABSTRACT FROM AUTHOR]

We continue building up the theory of complementarity spectra of connected graphs initiated by Fernandes et al. (Linear Algebra Appl 527:216–231, 2017). The complementarity spectrum of a connected graph G is the set Π (G) of complementarity eigenvalues of the adjacency matrix of G. We study what sort of information on G can be derived from the knowledge of Π (G) . We consider also the case in which Π (G) is partially known: it may happen, for instance, that only a few complementarity eigenvalues are available. Of special interest are the 4th and 5th smallest complementarity eigenvalues, so we give a close look at them. Finally, we consider the problem of estimating the gap between two consecutive complementarity eigenvalues. [ABSTRACT FROM AUTHOR]

Several techniques for extracting excited states from the Hamiltonian operator have been studied in recent years. This analysis is essential in areas such as Molecular Chemistry and other branches of Quantum Physics, especially those related to optical spectra and chemical reaction processes. Moreover, the knowledge of the operators' spectrum is required in other fields outside the Physics domains, since there are problems related to graph connectivity and principal component analysis which require a knowledge of, if not all, some important values of the operator spectrum. For those cases, part of the results clarifies that the eigenstate deflation can be used as a tool to extract the excited states. In consonance with these properties, we present a framework to perform the spectral decomposition for any dimension operator in a procedure based on the Householder reflections and unconstrained optimization using a meta-heuristic technique named Differential Evolution. The technique is discussed through examples and implemented in the superconducting real quantum device from IBM Quantum, whose results (without neglecting the system decoherence) follows the pattern expected for the respective models. We conclude our analysis, showing that the proposed framework presents encouraging results to exploit other scenarios in this field. [ABSTRACT FROM AUTHOR]

The fault-tolerant resolvability is an extension of metric resolvability in graphs with several intelligent systems applications, for example, network optimization, robot navigation, and sensor networking. The graphs of convex polytopes, which are rotationally symmetric, are essential in intelligent networks due to the uniform rate of data transformation to all nodes. A resolving set is an ordered set 핎 of vertices of a connected graph G in which the vector of distances to the vertices in 핎 uniquely determines all the vertices of the graph G. The minimum cardinality of a resolving set of G is known as the metric dimension of G. If 핎 ∖ ρ is also a resolving set for each ρ in 핎. In that case, 핎 is said to be a fault-tolerant resolving set. The fault-tolerant metric dimension of G is the minimum cardinality of such a set 핎. The metric dimension and the fault-tolerant metric dimension for three families of convex polytope graphs are studied. Our main results affirm that three families, as mentioned above, have constant fault-tolerant resolvability structures. [ABSTRACT FROM AUTHOR]

In this article, we consider the resistance matrix of a connected graph. A connected graph is said to be resistance regular if all the row(column) sums of its resistance matrix are equal. We establish some necessary and sufficient conditions for a simple connected graph to be a resistance regular graph. Also, we find some relationship between the Laplacian matrix and the resistance matrix in the case of weighted graphs where all edge weights are positive definite matrices of given order. [ABSTRACT FROM AUTHOR]

Let G = G(V,E) be a non-trivial simple connected graph. The length of the shortest path between two vertices p and q, represented by d(p; q), is called the distance between the vertices p and q. The distance between an edge ε = pq and a vertex r in G is defined as d(ε, r) = min{d(p, r), d(q, r)}. If d(r, p) ≠ d(r, q), then the vertex r is said to distinguish (resolve or recognize) two elements (edges or vertices) p, q ∈ V ∪ E. The minimum cardinality of a subset R (Re) of vertices such that all other vertices (edges) of the graph G are uniquely determined by their distances to the vertices in R (Re) is the metric dimension (edge metric dimension) of a graph G. In this article, we consider a family of pentagonal circular ladder (Pm) and investigate its edge metric dimension. We show that, for Pm the edge metric dimension is strictly greater than its metric dimension. Additionally, we answer a problem raised in the recent past, regarding the edge metric dimension of a family of a planar graph Rm (exists in the literature). [ABSTRACT FROM AUTHOR]

Let R be a ring with unity. In this paper, we introduce a new graph associated with R called the simple-intersection graph of R, denoted by GS(R). The vertices of GS(R) are the nonzero ideals of R, and two vertices are adjacent if and only if their intersection is a nonzero simple ideal. We study the interplay between the algebraic properties of R, and the graph properties of GS(R) such as connectedness, bipartiteness, girth, dominating sets, etc. Moreover, we determine the precise values of the girth and diameter of GS(R), as well as give a formula to compute the clique and domination numbers of GS(R).

The rings considered in this article are commutative with identity and unless otherwise specified, they are not integral domains. Let R be a ring. Let A(R) denote the set of all annihilating ideals of R and let us denote A(R)\{(0)} by A(R) ∗ . With any ring R, Alilou and Amjadi, in [The sum-annihilating essential ideal graph of a commutative ring, Commun. comb. optim. 1 (2) (2016), 117-135] introduced and investigated an undirected graph, denoted by AER, whose vertex set is A(R) ∗ and distinct vertices I, J are adjacent in AER if and only if AnnR(I) + AnnR(J) is an essential ideal of R. In this article, with any ring R, we associate an undirected graph denoted by G(R), whose vertex set is A(R) ∗ and distinct vertices I, J are adjacent in G(R) if and only if I + J is an essential ideal of R. The aim of this article is to investigate the interplay between the ring-theoretic properties of a ring R and the graph-theoretic properties of G(R). [ABSTRACT FROM AUTHOR]

Let G be a connected graph and ζ the set of all cliques in G. In this paper we introduce the concepts of unique (ζ, ζ)-eccentric clique graphs and self (ζ, ζ)-centered graphs. Certain standard classes of graphs are shown to be self (ζ, ζ)-centered, and we characterize unique (ζ, ζ)-eccentric clique graphs which are self (ζ, ζ)-centered. [ABSTRACT FROM AUTHOR]

For a given mapping f in the framework of different spaces, the fixed-point equations of the form f x = x can model several problems in different areas, such as differential equations, optimization, and computer science. In this work, the aim is to find the best proximity point and to prove its uniqueness on partial metric spaces where the symmetry condition is preserved for several types of contractive non-self mapping endowed with a graph. Our theorems generalize different results in the literature. In addition, we will illustrate the usability of our outcomes with some examples. The proposed model can be considered as a theoretical foundation for applications to real cases. [ABSTRACT FROM AUTHOR]

For a group G, we define the (directed and undirected) prime index elements graph of G. We study some interplay between the algebraic properties of a group and the graph theoretic properties of its (directed and undirected) prime index elements graphs. Also we describe some relationships of this graph with the power graph and the reduced power graph of a group. [ABSTRACT FROM AUTHOR]

Suppose that the zero-divisor graph of a commutative semi-group S, be a complete graph with an end vertex. In this paper, we determine the structure of the annihilator graph S and we show that if Z(S)= S, then the annihilator graph S is a disconnected graph.

A graph that is connected G(V,E) is a graph in which there is at least one path connecting every two vertices in G; otherwise, it is called a disconnected graph. Labels or values can be assigned to the vertices or edges of a graph. A vertex-labeled graph is one in which only the vertices are labeled, and an edges-labeled graph is one in which only edges are assigned values or labels. If both vertices and edges are labeled, the graph is referred to as total labeling. If given n vertices and m edges, numerous graphs can be made, either connected or disconnected. This study will be discussed the number of disconnected vertices labeled graphs of order seven containing no parallel edges and may contain loops. The results show that number of vertices labeled connected graph of order seven with no parallel edges is N(G7,m, g)l= 6,727×Cm6; while for 7≤g≤ 21, N(G7,m, g)l= kg C(m−(g−6))g−1, where k7 =30,160, k8 = 30,765, k9=21,000, k10 =28,364, k11= 26,880, k12=26,460 , k13 = 20,790, k14 =10,290, k15 = 8,022, k16 = 2,940, k17 =4,417, k18 = 2,835, k19 =210, k20 = 21, k21= 1.

The vertex-degree function index  H f (Γ) is defined as H f (Γ) = ∑ v ∈ V (Γ) f (d (v)) for a function f (x) defined on non-negative real numbers. In this paper, we determine the extremal graphs with the maximum (minimum) vertex degree function index in the set of all n-vertex chemical trees, trees, and connected graphs. We also present the Nordhaus–Gaddum-type results for H f (Γ) + H f (Γ ¯) and H f (Γ) · H f (Γ ¯) . [ABSTRACT FROM AUTHOR]

We analyse the "interplay" between analytical properties of a real function on a metric space, on the one hand, and topological properties of its graph, on the other. In particular, we study functions with closed, compact, connected, pathwise connected, or locally connected graphs, and we give nine conditions on the graph which are equivalent to the continuity of a function. A main emphasis is put on examples and counterexamples which illustrate how significant our hypotheses are, and how far sufficient conditions are from being necessary. [ABSTRACT FROM AUTHOR]

Given graphs X and Y with vertex sets V (X) and V (Y) of the same cardinality, the friends-and-strangers graph FS (X , Y) is the graph whose vertex set consists of all bijections σ : V (X) → V (Y) , where two bijections σ and σ ′ are adjacent if they agree everywhere except for two adjacent vertices a , b ∈ V (X) such that σ (a) and σ (b) are adjacent in Y. The most fundamental question that one can ask about these friends-and-strangers graphs is whether or not they are connected; we address this problem from two different perspectives. First, we address the case of "typical" X and Y by proving that if X and Y are independent Erdős-Rényi random graphs with n vertices and edge probability p , then the threshold probability guaranteeing the connectedness of FS (X , Y) with high probability is p = n − 1 / 2 + o (1). Second, we address the case of "extremal" X and Y by proving that the smallest minimum degree of the n -vertex graphs X and Y that guarantees the connectedness of FS (X , Y) is between 3 n / 5 + O (1) and 9 n / 14 + O (1). When X and Y are bipartite, a parity obstruction forces FS (X , Y) to be disconnected. In this bipartite setting, we prove analogous "typical" and "extremal" results concerning when FS (X , Y) has exactly 2 connected components; for the extremal question, we obtain a nearly exact result. [ABSTRACT FROM AUTHOR]

Let R be a ring with unity. In this paper, we introduce a new graph associated with R called the simple-intersection graph of R, denoted by GS (R). The vertices of GS (R) are the nonzero ideals of R, and two vertices are adjacent if and only if their intersection is a nonzero simple ideal. We study the interplay between the algebraic properties of R, and the graph properties of GS (R) such as connectedness, bipartiteness, girth, dominating sets, etc. Moreover, we determine the precise values of the girth and diameter of GS (R), as well as give a formula to compute the clique and domination numbers of GS (R). [ABSTRACT FROM AUTHOR]

A graph is supereulerian if it has a spanning eulerian subgraph. In this paper we construct a supereulerian graph controlling its matching number, minimum degree and edge connectivity, where δ(G) < α′(G) [ABSTRACT FROM AUTHOR]

Let Gn = (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 ∈ V are adjacent if and only if gcd(u, v) > 1. Connectedness, completeness, minimum degree, maximum degree, independence number, domination number and Eulerian property of the graph Gn are studied in this paper. [ABSTRACT FROM AUTHOR]

A molecular (chemical) graph is a simple connected graph, where the vertices represent the compound’s atoms and the edges represent bonds between the atoms, and the degree (valence) of every vertex (atom) is not more than four. In this paper, we determine the edge metric basis and edge metric dimension (EMD) of the complex molecular graph of a one-heptagonal carbon nanocone ( $HCN_{7}(q)$ ). We prove that only three non-adjacent vertices are the minimum requirement for the identification of all the edges in $HCN_{7}(q)$ , uniquely.

We use the finite dimensional monotonicity methods in order to investigate problems connected with the discrete s x , · -Laplacian on simple, connected, undirected, weighted, and finite graphs with nonlinearities given in a non-potential form. Positive solutions are also considered. [ABSTRACT FROM AUTHOR]

The irregularity of a graph G = (V, E) is defined as the sum of imbalances ∣du − dv∣ over all edges uv ∈ E, where du denotes the degree of the vertex u in G. This graph invariant, introduced by Albertson in 1997, is a measure of the defect of regularity of a graph. In this paper, we completely determine the extremal values of the irregularity of connected graphs with n vertices and p pendant vertices (1 ⩽ p ⩽ n − 1), and characterize the corresponding extremal graphs. [ABSTRACT FROM AUTHOR]

Given a graph with n points and m lines. If each vertex is labeled, then it can be constructed many graphs, connected, or disconnected graphs. A graph G is called a connected graph if there is at least one path that connects a pair of vertices in G. In addition, the graph formed may be simple or not simple. A simple graph is a graph that does not contain loops or parallel lines. A loop is a line that connects a point to itself, and a parallel line is two or more lines that connect the same pair of points. This paper will discuss the relationship between the formula patterns for calculating the number of connected graphs labeled with vertices of order five and six without loops.

Let \(G\) be an undirected simple graph on n vertices and \(\sigma^2(G)=n-2\) (degree sum of any two non-adjacent vertices in \(G\) is equal to \(n-2\)) and \(\alpha(G)\) be the cardinality of an maximum independent set of \(G\). In \(G\), a vertex of degree \((n-1)\) is called total vertex. We show that, for \(n\geq3\) is an odd number then \(\alpha(G)=2\) and \(G\) is a disconnected graph; for \(n\geq4\) is an even number then \(2\leq\alpha(G)\leq(n+2)/2\), where if \(\alpha(G)=2\) then \(G\) is a disconnected graph, otherwise \(G\) is a connected graph, \(G\) contains \(k\) total vertices and \(n-k\) vertices of degree \(\delta=(n-2)/2\), where \(0\leq k\leq(n-2)/2\). In particular, when \(k=0\) then \(G\) is an \((n-2)/2\)-Regular graph.