Related papers: Multiset and Mixed Metric Dimension for Starphene …
Introduction: molecular geometry, the three-dimensional arrangement of atoms within a molecule, is fundamental to understanding chemical reactivity, physical properties, and biological activity. The prevailing models used to describe…
A set of vertices $S$ resolves a graph if every vertex is uniquely determined by its vector of distances to the vertices in $S$. The metric dimension of a graph is the minimum cardinality of a resolving set of the graph. Fix a connected…
For a graph $G=(V,E)$, a set $S \subseteq V$ is a $[1,2]$-set if it is a dominating set for $G$ and each vertex $v \in V \setminus S$ is dominated by at most two vertices of $S$, i.e. $1 \leq \vert N(v) \cap S \vert \leq 2$. Moreover a set…
Molecular graphs of unsaturated carbon frameworks or hydrocarbons pruned of hydrogen atoms, are chemical graphs. A chemical graph is a connected simple graph of maximum degree $3$ or less. A nut graph is a connected simple graph with a…
It is of interest to study a wavelet system with a minimum number of generators. It has been showed by X. Dai, D. R. Larson, and D. M. Speegle in [11] that for any $d\times d$ real-valued expansive matrix M, a homogeneous orthonormal…
The metric dimension of a graph $G$ is the minimum number of vertices in a subset $S$ of the vertex set of $G$ such that all other vertices are uniquely determined by their distances to the vertices in $S$. In this paper we investigate the…
Given a connected graph $G$, a vertex $w\in V(G)$ distinguishes two different vertices $u,v$ of $G$ if the distances between $w$ and $u$ and between $w$ and $v$ are different. Moreover, $w$ strongly resolves the pair $u,v$ if there exists…
Nonlocal metric dimension ${\rm dim}_{\rm n\ell}(G)$ of a graph $G$ is introduced as the cardinality of a smallest nonlocal resolving set, that is, a set of vertices which resolves each pair of non-adjacent vertices of $G$. Graphs $G$ with…
Let $G$ be a group. The intersection graph $\Gamma(G)$ of $G$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper non-trivial subgroups of $G$, and there is an edge between two…
Monatomic metal (e.g. silver) structures could form preferably at graphene edges. We explore their structural and electronic properties by performing density functional theory based first-principles calculations. The results show that…
RNA molecules are known to form complex secondary structures including pseudoknots. A systematic framework for the enumeration, classification and prediction of secondary structures is critical to determine the biological significance of…
A matching $M$ in a graph $\Gamma$ is positive if $\Gamma$ has a vertex-labeling such that $M$ coincides with the set of edges with positive weights. A positive matching decomposition (pmd) of $\Gamma$ is an edge-partition $M_1,\ldots,M_p$…
In this paper we characterize a mathematical model called Maximum Common Subelement (MCS) Model and prove the existence of four different metrics on such model. We generalize metrics on graphs previously proposed in the literature and…
The metric dimension of a graph $G$ is the size of a smallest subset $L \subseteq V(G)$ such that for any $x,y \in V(G)$ with $x\not= y$ there is a $z \in L$ such that the graph distance between $x$ and $z$ differs from the graph distance…
The median of a graph $G$ with weighted vertices is the set of all vertices $x$ minimizing the sum of weighted distances from $x$ to the vertices of $G$. For any integer $p\ge 2$, we characterize the graphs in which, with respect to any…
For an ordered set W = {w1,w2,...,wk} of vertices and a vertex v in a connected graph G, the ordered k-vector r(v|W) := (d(v,w1),d(v,w2),...,d(v,wk)) is called the (metric) representation of v with respect to W, where d(x,y) is the distance…
Let $G=(V,E)$ be a simple connected graph. A matching $M$ in a graph $G$ is a collection of edges of $G$ such that no two edges from $M$ share a vertex. A matching $M$ is maximal if it cannot be extended to a larger matching in $G$. The…
The undirected zero divisor graph of a commutative ring with unity \( R \), denoted by \( \Gamma(R) = (V(\Gamma(R)), E(\Gamma(R))) \). The vertex set \( V(\Gamma(R)) \) consists of all the non-zero zero-divisors of \( R \). The edge set \(…
A great efficacy of molecular quantum chemistry applied to basic graphene problems has been recently demonstrated by the authors when studying the formation of peculiar composites between carbon nanotubes and graphene as well as considering…
This article investigates multiset dimensions in zero divisor graphs (ZD-graphs) associated with rings. Through rigorous analysis, we establish general bounds for the multiset dimension (Mdim) in ZD-graphs, exploring various commutative…