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Let $ \Gamma =(V,E) $ Γ=(V,E)be a connected graph of order n. Two vertices pand qin Vare said to resolve by a vertex $ x \in V $ x∈Vif $ d(p,x)\neq d(q,x) $ d(p,x)≠d(q,x). An ordered subset $ F=\{k_{1},k_{2},k_{3},\dots,k_{l}\} $ F={k1,k2,k3,…,kl}of vertices in Γ is said to be resolving set if for every pair p, qof distinct vertices in V, we have $ \zeta (p|F)\neq \zeta (q|F) $ ζ(p|F)≠ζ(q|F), where $ \zeta (a|F)=(d(a,k_{1}),d(a,k_{2}),d(a,k_{3}),\dots,d(a,k_{l})) $ ζ(a|F)=(d(a,k1),d(a,k2),d(a,k3),…,d(a,kl))is the l-code/metric coordinate representation of the vertex awith respect to the set F. The resolving set for Γ with minimum cardinality is known as metric basis for the graph Γ and the cardinality of metric basis is called as metric dimension of Γ. In this work, we demonstrate that for two families of convex polytopes which are closely linked, the metric dimension is constant.

ABSTRACTIn this work, we introduce the concept of independence in multiresolving sets and present some basic results on it. We characterize two graph families for which we prove that their multiset dimension is three. We also provide a corrected proof for the multiset dimension of the grid graph , due to Simanjuntak et al., present in the literature.

Polycyclic aromatic hydrocarbons (PAHs) are a class of persistent organic contaminants that contain two or more aromatic rings. PAHs are commonly found in soil, water, and air, and result in mainly incomplete combustion of carbonaceous products. PAHs have a wide range of applications, such as interstellar species, optical and electrical materials, pharmaceutical industry, and other functional materials in petrochemistry. The chemical graph theory helps in understanding the complex structure of molecules. The parameters of resolvability for graph Γ=(V,E)are a relatively novel advanced area where the entire structure is constructed so that each vertex (atom) or/and edge (bond) represents a unique position. In this paper, we study the resolvability parameter, i.e., mixed metric dimension for the complex molecular structure of PAHs. We also prove that the mixed metric dimension is unbounded and is not constant for PAHs.

Let be a simple, connected, and undirected graph. The distance between two vertices denoted by , is the length of the shortest path connecting uand v. A subset of vertices is said to be a resolving set for Gif for any two distinct vertices V, there exist a vertex such that . A minimal resolving set is called a metric basis, and the cardinality of the basis set is called the metric dimension of G, denoted by . In this article, we find the metric dimension for two infinite families of plane graphs and , where is obtained by the combination of copies of bipartite graphs , and is obtained by the combination of double antiprism graph with antiprism graph and then adding n-paths of length p.

AbstractLet Rbe a commutative ring with identity. The zero-divisor graph of Rdenoted by is an undirected graph where is the set of non-zero zero-divisors of Rand there is an edge between the vertices z1and z2in if A set of vertices S resolves a graph Gif every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of Gis the minimum cardinality of a resolving set of G. If we remove any vertex in a resolving set, then the resulting set is also a resolving set, called the fault-tolerant resolving set, and its minimum cardinality is called the fault-tolerant metric dimension. In this article, we study the fault-tolerant metric dimension for where R= and Furthermore, we obtain some results regarding the line graph of and the zero-divisor graph of a Cartesian product of fields.

Let R be a Noetherian integral domain which is also an algebra over ℚ (ℚ is the field of rational numbers). Let σ be an endo-morphism of R and δ a σ-derivation of R. We recall that a ring R is a weak (σ, δ)-rigid ring if a(σ(a)+ δ(a)) ∈N(R) if and only if a ∈N(R) for a ∈R (N(R) is the set of nilpotent elements of R). With this we prove that if R is a Noetherian integral domain which is also an algebra over ℚ, σ an automorphism of R and δ a σ-derivation of R such that R is a weak (σ, δ)-rigid ring, then N(R) is completely semiprime.

Metric basis and metric dimension has become an integral part of molecular topology and combinatorial chemistry. It has a lot of applications in pharmacy, chemistry, computer, and mathematical disciplines. Let Φ=(V,E)be a simple connected graph. A subset F={y1,y2,y3,…,ys}of distinct vertices from V(Φ)is called a resolving set if for any a∈V(Φ), the metric code of awith respect to Fthat is denoted by ζ(a|F), which is defined as ζ(a|F)=(d(y1,a),d(y2,a),d(y3,a),…,d(ys,a)), is distinct for distinct a. Then, such a set Fwith minimum cardinality is called the metric basis for Φ, and this minimum cardinality of a resolving set Fis called the metric dimension of Φand is denoted by dim(Φ). A polytope in elementary geometry is a geometric object with flat sides. The polytopes which are convex sets and are contained in the n-dimensional space Rn(Euclidean space) are termed convex polytopes. Convex polytopes have found a lot of applications in different areas of computer science and mathematics. In this work, we construct two closely related families of convex polytope graphs (viz., Dnand Qn), using the existing families of convex polytope graphs. We study the metric dimension for these graphs and prove that only three vertices are a minimum requirement for the unique identification of all vertices in these graphs. We also give some partial answers to the problems raised in the recent past.