Related papers: Pebbling number of polymers

Let $G=(V,E)$ be a graph and $e=uv\in E$. Define $n_u(e,G)$ be the number of vertices of $G$ closer to $u$ than to $v$. The number $n_v(e,G)$ can be defined in an analogous way. The Mostar index of $G$ is a new graph invariant defined as…

Let $G=(V,E)$ be a simple undirected graph with $n$ vertices then a set partition $\pi=\{V_1, ..., V_k\}$ of the vertex set of $G$ is a connected set partition if each subgraph $G[V_j]$ induced by the blocks $V_j$ of $\pi$ is connected for…

The distinguishing number $D(G)$ of a graph $G$ is the least integer $d$ such that $G$ has a vertex labeling with $d$ labels that is preserved only by a trivial automorphism. The distinguishing stability, of a graph $G$ is denoted by…

The deck of a graph $G$ is given by the multiset of (unlabelled) subgraphs $\{G-v:v\in V(G)\}$. The subgraphs $G-v$ are referred to as the cards of $G$. Brown and Fenner recently showed that, for $n\geq29$, the number of edges of a graph…

A graph composition is a partition of the vertex set such that each member of the partition induces a connected sub- graph, and the composition number of a graph is the number of possible graph compositions. A partition of a set S of…

For $G$ a simple, connected graph, a vertex labeling $f:V(G)\rightarrow \mathbb{Z}_+$ is called a $\textit{radio labeling of}$ $G$ if it satisfies $|f(u)-f(v)|\geq \operatorname{diam}(G) + 1 - d(u,v)$ for all distinct vertices $u,v\in…

$f$-vertex stability number $vs_f(G)=\min\{|X|: X\subseteq V(G) \enspace \text{and} \enspace f(G-X)\neq f(G)\}$, and $f$-edge stability number is defined similarly by setting $X\subseteq E(G)$. In this paper, for multiplicative and mining…

Let $F_k$ be the set of graphs on $k$ vertices. For a graph $G$, a $k$-decomposition is a set of induced subgraphs of $G$, each isomorphic to an element of $F_k$, such that each pair of vertices of $G$ is in exactly one element of the set.…

A scramble on a connected multigraph is a collection of connected subgraphs that generalizes the notion of a bramble. The maximum order of a scramble, called the scramble number of a graph, was recently developed as a tool for lower…

Given a connected graph $G=(V(G), E(G))$, the length of a shortest path from a vertex $u$ to a vertex $v$ is denoted by $d(u,v)$. For a proper subset $W$ of $V(G)$, let $m(W)$ be the maximum value of $d(u,v)$ as $u$ ranging over $W$ and $v$…

A 1-selection $f$ of a graph $G$ is a function $f: V(G)\rightarrow E(G)$ such that $f(v)$ is incident to $v$ for every vertex $v$. The 1-removed $G_f$ is the graph $(V(G),E(G)\setminus f[V(G)])$. The (1-)robust chromatic number $\chi_1(G)$…

Let $G$ be a finite group, and let ${\rm{cd}}(G)$ denote the set of degrees of the irreducible complex characters of $G$. The degree graph $\Delta(G)$ of $G$ is defined as the simple undirected graph whose vertex set ${\rm{V}}(G)$ consists…

Let $G$ be a finite, connected graph of order of at least 2, with vertex set $V(G)$ and edge set $E (G)$. A set $S$ of vertices of the graph $G$ is a doubly resolving set for $G$ if every two distinct vertices of $G$ are doubly resolved by…

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, and $L(G)$ be the line graph of $G$, which has vertex set $E(G)$ and two vertices $e$ and $f$ of $L(G)$ is adjacent if $e$ and $f$ is incident in $G$. The vertex-edge graph…

The fixing number of a graph $G$ is the smallest cardinality of a set of vertices $S$ such that only the trivial automorphism of $G$ fixes every vertex in $S$. The fixing set of a group $\Gamma$ is the set of all fixing numbers of finite…

Here we merge the two fields of Cops and Robbers and Graph Pebbling to introduce the new topic of Cops and Robbers Pebbling. Both paradigms can be described by moving tokens (the cops) along the edges of a graph to capture a special token…

Given a graph $G$, a configuration space of $G$ can be thought of as the set of all possible configurations of "robots" which can move throughout $G$, subject to some constraints. We introduce a type of configuration space which we call…

For any graph $G$ of order $p$, a bijection $f: V(G)\to [1,p]$ is called a numbering of the graph $G$ of order $p$. The strength $str_f(G)$ of a numbering $f: V(G)\to [1,p]$ of $G$ is defined by $str_f(G) = \max\{f(u)+f(v)\; |\; uv\in…

Let $G_{1}$ and $G_{2}$ be disjoint copies of a graph $G$, and let $g:V(G_{1})\rightarrow V(G_{2})$ be a function. A functigraph $F_{G}$ consists of the vertex set $V(G_{1})\cup V(G_{2})$ and the edge set $E(G_{1})\cup E(G_{2})\cup…

Let $r \geq 2$ be a real number. A complex nowhere-zero $r$-flow on a graph $G$ is an orientation of $G$ together with an assignment $\varphi\colon E(G)\to \mathbb{C}$ such that, for all $e \in E(G)$, the modulus of the complex number…