Distinct Vertices Research Articles

Bounds on radio mean number of graphs

A radio mean labeling l of G maps distinct vertices of G to distinct elements of ℤ + satisfying the radio mean condition that diam ( G ) + 1 - d G ( w , w ′ ) ≤ ⌈ l ( w ) + l ( w ′ ) 2 ⌉ , ∀ w , w ′ ∈ V ( G ) where dG (w, w′) is the smallest length of a w, w′- path in G and diam (G) = max {dG (w, w′) : w, w′ ∈ V (G)} is the diameter of G. The radio mean number of graph G is defined as rmn (G) = min {span (l) : l isaradiomeanlabelingof G} where span(l) is given by max {l (w) : w ∈ V (G)}. It has been proved in literature that |V (G) | ≤ rmn (G) ≤ |V (G) | + diam (G) -2. Cryptographic algorithms can exploit the unique radio mean number associated with a graph to generate keys. An exhaustive listing of all feasible radio mean labelings and their span is essential to obtain the radio mean number of a given graph. Since the radio mean condition depends on the distance between vertices and the graph’s diameter, as the order and diameter increase, finding a radio mean labeling itself is quite difficult and, so is obtaining the radio mean number of a given graph. Here we discuss the extreme values of the radio mean number of a given graph of order n. In this article we obtained bounds on the radio mean number of a graph G of order n and diameter d in terms of the radio mean number of its induced subgraph H where diam (H) = d and dH (w, w′) = dG (w, w′) for any w, w′ ∈ V (H). The diametral path Pd+1 is one such induced subgraph of G and hence we have deduced the limits of rmn (G) in terms of rmn (Pd+1). It is known that if d = 1, 2 or 3, then rmn (G) = n. Here, we have given alternative proof for the same. The authors of this article have studied radio mean labeling of paths in another article. Using those results, we have improved the bounds on the radio mean number of a graph of order n and diameter d ≥ 4. It is also shown that among all connected graphs on n vertices, the path Pn of order n possesses the maximum radio mean number. This is the first article that has completely solved the question of maximum and minimum attainable radio mean numbers of graphs of order n.

Strong Resolving Hop Domination in Graphs

A vertex w in a connected graph G strongly resolves two distinct vertices u and v in V (G) if v is in any shortest u-w path or if u is in any shortest v-w path. A set W of vertices in G is a strong resolving set G if every two vertices of G are strongly resolved by some vertex of W. A set S subset of V (G) is a strong resolving hop dominating set of G if S is a strong resolving set in G and for every vertex v ∈ V (G) \ S there exists u ∈ S such that dG(u, v) = 2. The smallest cardinality of such a set S is called the strong resolving hop domination number of G. This paper presents the characterization of the strong resolving hop dominating sets in the join, corona and lexicographic product of graphs. Furthermore, this paper determines the exact value or bounds of their corresponding strong resolving hop domination number.

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.

Cliques and Supercliques in Graphs

A set S ⊆ V (G) of an undirected graph G is a clique if every two distinct vertices in S are adjacent. A clique is a superclique if for every pair of distinct vertices v, w ∈ S, there exists u ∈ V (G) \ S such that u ∈ NG(v) \ NG(w) or u ∈ NG(w) \ NG(v). The maximum cardinality of a clique (resp. superclique) in G is called the clique (resp. superclique) number of G. In this paper, we determine the clique and superclique numbers of some graphs.

Stable Locating-Dominating Sets in the Edge Corona and Lexicographic Product of Graphs

A set S ⊆ V (G) of an undirected graph G is a locating-dominating set of G if for each v ∈ V (G) \ S, there exists w ∈ S such tha vw ∈ E(G) and NG(x) ∩ S ̸= NG(y) ∩ S for any two distinct vertices x and y in V (G) \ S. S is a stable locating-dominating set of G if it is a locating-dominating set of G and S \ {v} is a locating-dominating set of G for each v ∈ S. The minimum cardinality of a stable locating-dominating set of G, denoted by γSLD(G), is called the stable locating-domination number of G. In this paper, we investigate this concept and the corresponding parameter for edge corona and lexicographic product of graphs.

Parameterised temporal exploration problems

We study the fixed-parameter tractability of the problem of deciding whether a given temporal graph admits a temporal walk that visits all vertices (temporal exploration) or, in some variants, a certain subset of the vertices. In the strict variant, edges must be traversed in strictly increasing timesteps; in the non-strict variant, any number of edges can be traversed in each timestep. For both variants, we give FPT algorithms for finding a temporal walk that visits a given set X of vertices, parameterised by |X|, and for finding a temporal walk that visits at least k distinct vertices, parameterised by k. We also show W[2]-hardness for a set version of temporal exploration. For the non-strict variant, we give an FPT algorithm for temporal exploration parameterised by the lifetime, and show that temporal exploration can be solved in polynomial time if the graph in each timestep has at most two connected components.

A vertex-separator-based integer linear programming formulation for the partitioned Steiner tree problem

Given an undirected graph G with a cost function on vertices, a collection of subgraphs of G such that in each subgraph, there are some distinguished vertices called terminals, the Partitioned Steiner Tree Problem (PSTP) asks for a minimum cost vertex set such that, in each of the given subgraph Gi, the graph induced by the vertex set spans the terminal set in Gi. The PSTP generalizes the well-known Steiner tree problem and has important applications in computational sustainability, network design, and social network analysis. However, for solving the PSTP, conventional integer programming approaches based on single-commodity flow, multi-commodity flow and subtour elimination integer linear programs, suffer from low computational efficiency due to a substantial number of variables. In this paper, we propose a compact vertex-separator-based integer linear programming formulation with much fewer variables. Enhancing inequalities are also studied for tightening the formulation. We further investigate a branch-and-cut algorithm, a local-branching heuristic algorithm, and a hybrid algorithm combining them. In experiments where both public real-world and synthetic graphs are used, our hybrid algorithm outperforms all conventional approaches, especially for large graphs with more than ten thousand vertices. Further tests also validate the effectiveness of the proposed formulation and enhancing inequalities.

On the ideal-based zero-divisor graph of commutative rings

Let [Formula: see text] be a finite commutative ring with identity, [Formula: see text] be an ideal of [Formula: see text] and [Formula: see text] denotes the Jacobson radical of [Formula: see text]. The ideal-based zero-divisor graph [Formula: see text] of [Formula: see text] is a graph with vertex set [Formula: see text] for some [Formula: see text] in which distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we determine the diameter, girth of [Formula: see text]. Specifically, we classify all finite commutative nonlocal rings for which [Formula: see text] is perfect. Furthermore, we discuss about the planarity, outerplanarity, genus and crosscap of [Formula: see text] and characterize all of them.

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 $\mathbb{A}\subseteq S^*(\mathbb{V})=S(\mathbb{V})\backslash\{0\}.$

Planar index and outerplanar index of zero-divisor graphs of commutative rings without identity

Let $R$ be a commutative ring without identity. The zero-divisor graph of $R,$ denoted by $\Gamma(R)$ is a graph with vertex set $Z(R)\setminus \{0\}$ which is the set of all nonzero zero-divisor elements of $R,$ and two distinct vertices $x$ and $y$ are adjacent if and only if $xy=0.$ In this paper, we characterize the rings whose zero-divisor graphs are ring graphs and outerplanar graphs. Further, we establish the planar index, ring index and outerplanar index of the zero-divisor graphs of finite commutative rings without identity.

Wiener index and addressing of some finite graphs

An addressing of length t of a graph G is an assignment of t-tuples with entries in called addresses, to the vertices of G such that the distance between any two vertices can be determined from their addresses. Let Z(R) be the set of zero-divisors of a commutative ring R. In this article, we investigate the adjacency matrix of a simple graph whose vertex set is and two distinct vertices x and y are adjacent if and only if where are distinct primes greater than two and α, β are positive integers. Moreover, we estimate addressing of and obtain Wiener and Zagreb indices and some energies of it.

Calculating different indices of Idempotent graph of ring ℤn

By Idempotent graph of ℤn we mean the graph whose vertices are all elements of ℤn such that distinct vertices u and v are adjacent if and only if u + v is an idempotent element of ℤn, denoted by Gid(ℤn). In this paper, we compute some kind of topological indices of Idempotent graph of ℤn.

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.

Mixed integer programming and quadratic programming formulations for the interval count problem

A graph is an interval graph if its vertex set corresponds to a family of intervals on the real line, called a model, such that two distinct vertices are adjacent in the graph if and only if their corresponding intervals intersect each other. The minimum number of interval lengths that suffices to represent a model of a given interval graph is its interval count. The use of mathematical optimization techniques for solving interval count problems was first explored by Joos et al.[1]. In more detail, given a bipartition of vertices into classes of lengths, the authors propose an efficient linear programming based algorithm for solving the interval count two problem. However, so far, no mathematical formulation exists in the literature for general interval count. As a contribution in that direction, we introduce a mixed integer programming formulation for the exact value of interval count, parameterized by the largest interval length. Additionally, we also propose a quadratic formulation for a valid upper bound on interval count. Solution algorithms for these formulations were tested on interval count instances found in the literature. As an outcome of these experiments, the algorithm for the upper bound formulation was shown to run much faster than its exact solution counterpart. Furthermore, the upper bounds thus obtained were frequently certified as optimal by the exact algorithm.

Counting lines in semi-complete digraphs *

A digraph D= (V, A) is semi-complete if for each pair of distinct vertices x and y in V, either xy or yx belong to A. A subset ℓ of vertices is a line of D if there are two distinct vertices x and y such that for any vertex zε V, zε ℓ if and only if a directed shortest path exists containing x,y and z.A classic result proved by Erdös says that any set of n points in the Euclidean plane endowed with the Euclidean distance defines a metric space with at least n different lines unless there is a line containing the n points. Chen and Chvátal in 2008 conjectured that the same results is true for any metric spaces where lines are defined in a manner similar to above.In this paper we prove that in any semi-complete digraphs with n vertices the number of lines defined by vertices connected by an arc is at least n. Then, the quasi-metric spaces defined by semi-complete digraphs fulfill Chen and Chvátal conjecture in a stronger manner as, on the one hand, they always have at least n lines, and on the other hand, these n lines are defined by vertices at distance one.

On clique numbers of colored mixed graphs

An (m,n)-colored mixed graph, or simply, an (m,n)-graph is a graph having m different types of arcs and n different types of edges. A homomorphism of an (m,n)-graph G to another (m,n)-graph H is a vertex mapping that preserves adjacency, the type thereto and the direction. A subset R of the set of vertices of G that always maps distinct vertices in itself to distinct image vertices under any homomorphism is called an (m,n)-relative clique of G. The maximum cardinality of an (m,n)-relative clique of a graph is called the (m,n)-relative clique number of the graph. In this article, we explore the (m,n)-relative clique numbers for various families of graphs.

On the Relation of the Total Graph of a Ring and a Product of Graphs

The total graph of a ring R, denoted as T(Γ(R)), is defined to be a graph with vertex set V(T(Γ(R)))=R and two distinct vertices u,v∈V(T(Γ(R))) are adjacent if and only if u+v∈Z(R), where Z(R) is the zero divisor of R. The Cartesian product of two graphs G and H is a graph with the vertex set V(G×H)=V(G)×V(H) and two distinct vertices (u_1,v_1 ) and (u_2,v_2 ) are adjacent if and only if: 1) u_1=u_2 and v_1 v_2∈H; or 2) v_1=v_2 and u_1 u_2∈E(G). An isomorphism of graphs G dan H is a bijection ϕ:V(G)→V(H) such that u,v∈V(G) are adjacent if and only if f(u),f(v)∈V(H) are adjacent. This paper proved that T(Γ(Z_2p )) and P_2×K_p are isomorphic for every odd prime p.

Indeks Eksentrisitas Zagreb Pertama dan Kedua Graf Koprima dari Grup Matriks Upper Unitriangular atas Ring Bilangan Bulat Modulo Prima

The coprime graph of a group G is a graph Γ_G with G is its set of vertices and any two distinct vertices are adjacent if and only if their order are relatively prime. Let p be a prime number, then G_p denotes the multiplicative group of 2×2 upper unitriangular matrices over ring of integers modulo p. The purposes of this research are to study the coprime graph Γ_(G_p ) and find the first and the second Zagreb eccentricity indices of Γ_(G_p ) for p≥3. The results of this research are as follows. First Zagreb eccentricity index of coprime graph Γ_(G_p )isE_1 (Γ_(G_p ))=4p-3. Second Zagreb eccentricity index of coprime graph Γ_(G_p )isE_2 (Γ_(G_p ))=2p-2.

Maximum and Minimum Degree Energy of Commuting Graph for Dihedral Groups

If G is a finite group and Z(G) is the centre of G, then the commuting graph for G, denoted by ΓG, has G\Z(G) as its vertices set with two distinct vertices vp and vq are adjacent if vp vq = vq vp. The degree of the vertex vp of ΓG, denoted by 𝑑𝑑𝑣𝑣𝑝𝑝 , is the number of vertices adjacent to vp. The maximum (or minimum) degree matrix of ΓG is a square matrix whose (p,q)-th entry is max{𝑑𝑑𝑣𝑣𝑝𝑝,𝑑𝑑𝑣𝑣𝑞𝑞 } (or min{𝑑𝑑𝑣𝑣𝑝𝑝,𝑑𝑑𝑣𝑣𝑞𝑞 }) whenever vp and vq are adjacent, otherwise, it is zero. This study presents the maximum and minimum degree energies of ΓG for dihedral groups of order 2n, D2n by using the absolute eigenvalues of the corresponding maximum degree matrices (MaxD(ΓG)) and minimum degree matrices (MinD(ΓG)). Here, the comparison of maximum and minimum degree energy of ΓG for D2n is discussed by considering odd and even n cases. The result shows that for each case, both energies are non-negative even integers and always equal.