Let $d(u,v)$ denote the distance between two distinct vertices of a connected graph $G$, and $\diam(G)$ be the diameter of $G$. A radio labeling $c$ of $G$ is an assignment of positive integers to the vertices of $G$ satisfying $d(u,v)+|c(u)-c(v)|\geq \diam(G) + 1.$ The maximum integer in the range of the labeling is its span. The radio number of $G$, $rn(G)$, is the minimum possible span. The family of gear graphs of order $n$, $G_n$, consists of planar graphs with $2n+1$ vertices and $3n$ edges. We prove that the radio number of the $n$-gear is $4n+2$.
Distance matrices of some star like graphs are investigated in \cite{JAK}. These graphs are trees which are stars, wheel graphs, helm graphs and gear graphs. Except for gear graphs in the above list of star like graphs, there are precise formulas available in the literature to compute the inverse/Moore-Penrose inverse of their distance matrices. These formulas tell that if $D$ is the distance matrix of $G$, then $D^\dagger = -\frac{1}{2}L+uu'$, where $L$ is a Laplacian-like matrix which is positive semidefinite and all row sums equal to zero. The matrix $L$ and the vector $u$ depend only on th…
The zero forcing numbers and propagation times of gear graphs and helm graphs
University of North Texas（北德克萨斯大学）；Rice University（莱斯大学）；Trinity College（三一学院）；University of Denver（丹佛大学）；Southeast Missouri State University（密苏里东南州立大学）
We introduce a new method for constructing isospectral quantum graphs that is based on transplanting derivatives of eigenfunctions. We also present simple digraphs with the same reversing zeta function, which generalizes the Bartholdi zeta function to digraphs.
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 $u,v\in V(G)$. The $\textit{radio number}$ of $G$ is the minimal span over all radio labelings of $G$. If a bijective radio labeling onto $\{1,2,...,|V(G)|\}$ exists, $G$ is called a $\textit{radio graceful graph}$. We determine the radio number of all diameter $3$ Hamming graphs and show that an infinite subset of them is radio graceful.
A radio labeling of a connected graph $G$ is a function $c:V(G) \to \mathbb Z_+$ such that for every two distinct vertices $u$ and $v$ of $G$ $$\text{distance}(u,v)+|c(u)-c(v)|\geq 1+ \text{diameter}(G).$$ The radio number of a graph $G$ is the smallest integer $M$ for which there exists a labeling $c$ with $c(v)\leq M$ for all $v\in V(G)$. The radio number of graphs of order $n$ and diameter $n-1$, i.e., paths, was determined in \cite{paths}. Here we determine the radio numbers of all graphs of order $n$ and diameter $n-2$.
For $k\in\mathbb{Z}^+$ and $G$ a simple connected graph, a $k$-radio labeling $f:V_G\to\Z^+$ of $G$ requires all pairs of distinct vertices $u$ and $v$ to satisfy $|f(u)-f(v)|\geq k+1-d(u,v)$. When $k=1$, this requirement gives rise to the familiar labeling known as vertex coloring for which each vertex of a graph is labeled so that adjacent vertices have different "colors". We consider $k$-radio labelings of $G$ when $k=\diam(G)$. In this setting, no two vertices can have the same label, so graphs that have radio labelings of consecutive integers are one extreme on the spectrum of possibiliti…
For a connected graph $G$, let $diam(G)$ and $d(u,v)$ denote the diameter of $G$ and distance between $u$ and $v$ in $G$. A radio labeling of a graph $G$ is a mapping $φ: V(G) \rightarrow \{0,1,2,...\}$ such that $|φ(u)-φ(v)| \geq diam(G) + 1 - d(u,v)$ for every pair of distinct vertices $u, v$ of $G$. The span of $φ$ is defined as span($φ$) = $\max\{|φ(u)-φ(v)| : u, v \in V(G)\}$. The radio number $rn(G)$ of $G$ is defined as $rn(G)$ = $\min\{$span($φ$) : $φ$ is a radio labeling of $G\}$. In this paper, we determine the radio number for middle graph of paths.
A radio $k$-labeling of a connected graph $G$ is an assignment $c$ of non negative integers to the vertices of $G$ such that $$|c(x) - c(y)| \geq k+1 - d(x,y),$$ for any two vertices $x$ and $y$, $x\ne y$, where $d(x,y)$ is the distance between $x$ and $y$ in $G$. In this paper, we study radio labelings of distance graphs, i.e., graphs with the set $\Z$ of integers as vertex set and in which two distinct vertices $i, j \in \Z$ are adjacent if and only if $|i - j| \in D$.
The radio number problem uses a graph-theoretical model to simulate optimal frequency assignments on wireless networks. A radio labeling of a connected graph $G$ is a function $f:V(G) \to \mathbb Z_{0}^+$ such that for every pair of vertices $u,v \in V(G)$, we have $\lvert f(u)-f(v)\rvert \ge \text{diam}(G) + 1 - d(u,v)$ where $\text{diam}(G)$ denotes the diameter of $G$ and $d(u,v)$ the distance between vertices $u$ and $v$. Let $\text{span}(f)$ be the difference between the greatest label and least label assigned to $V(G)$. Then, the \textit{radio number} of a graph $\text{rn}(G)$ is defined…
For integers $k, r > 0$, a conditional $(k,r)$-coloring of a graph $G$ is a proper $k$-coloring of the vertices of $G$ such that every vertex $v$ of degree $d(v)$ in $G$ is adjacent to at least $\min\{r, d(v)\}$ differently colored vertices. Given $r$, the smallest integer $k$ for which $G$ has a conditional $(k,r)$-coloring is called the $r$th order conditional chromatic number $χ_r(G)$ of $G$. We give results (exact values or bounds for $χ_r(G)$, depending on $r$) related to the conditional coloring of some graphs. We introduce \emph{unique conditional colorability} and give some related res…
A Stacked-book graph Gm,n is obtained from the Cartesian product of a star graph Sm and a path Pn, where m and s are the orders of the star graph and the path respectively. Obtaining the radio number of a graph is a rigorous process, which is dependent on the diameter of G and positive difference of non-negative integer labels f(u) and f(v) assigned to any two u, v in the vertex set V (G) of G. This paper obtains tight upper and lower bounds of the radio number of Gm,n where the path Pn has an odd order. The case where Pn has an even order has been investigated.
For a graph $G$, a $k$-radio labeling of $G$ is the assignment of positive integers to the vertices of $G$ such that the closer two vertices are on the graph, the greater the difference in labels is required to be. Specifically, $\vert f(u)-f(v)\vert\geq k + 1 - d(u,v)$ where $f(u)$ is the label on a vertex $u$ in $G$. Here, we consider the case when $G$ is the Cartesian products of complete graphs. Specifically we wish to find optimal labelings that use consecutive integers and determine when this is possible. We build off of a paper by Amanda Niedzialomski and construct a framework for disco…
A radio labeling of a graph $G$ is a mapping $\vp : V(G) \rightarrow \{0, 1, 2,...\}$ such that $|\vp(u)-\vp(v)|\geq \diam(G) + 1 - d(u,v)$ for every pair of distinct vertices $u,v$ of $G$, where $\diam(G)$ and $d(u,v)$ are the diameter of $G$ and distance between $u$ and $v$ in $G$, respectively. The radio number $\rn(G)$ of $G$ is the smallest number $k$ such that $G$ has radio labeling with $\max\{\vp(v):v \in V(G)\}$ = $k$. In this paper, we slightly improve the lower bound for the radio number of graphs given by Das \emph{et al.} in [5] and, give necessary and sufficient condition to achi…
A radio labeling is an assignment $c:V(G) \rightarrow \textbf{N}$ such that every distinct pair of vertices $u,v$ satisfies the inequality $d(u,v)+|c(u)-c(v)|\geq \diam(G)+1$. The span of a radio labeling is the maximum value. The radio number of $G$, $rn(G)$, is the minimum span over all radio labelings of $G$. Generalized prism graphs, denoted $Z_{n,s}$, $s \geq 1$, $n\geq s$, have vertex set $\{(i,j)\,|\, i=1,2 \text{and} j=1,...,n\}$ and edge set $\{((i,j),(i,j \pm 1))\} \cup \{((1,i),(2,i+σ))\,|\,σ=-\left\lfloor\frac{s-1}{2}\right\rfloor\,\ldots,0,\ldots,\left\lfloor\frac{s}{2}\right\rflo…
A Stacked-book graph $G_{m,n}$ results from the Cartesian product of a star graph $S_m$ and path $P_n$, where $m$ and $n$ are the orders of $S_m$ and $P_n$ respectively. A radio labeling problem of a simple and connected graph, $G$, involves a non-negative integer function $f:V(G)\rightarrow \mathbb Z^+$ on the vertex set $V(G)$ of G, such that for all $u,v \in V(G)$, $|f(u)-f(v)| \geq \textmd{diam}(G)+1-d(u,v)$, where $\textmd {diam}(G)$ is the diameter of $G$ and $d(u,v)$ is the shortest distance between $u$ and $v$. Suppose that $f_{min}$ and $f_{max}$ are the respective least and largest v…
Radio labeling is a variation of Hale's channel assignment problem, in which one seeks to assign positive integers to the vertices of a graph $G$ subject to certain constraints involving the distances between the vertices. Specifically, a radio labeling of a connected graph $G$ is a function $c:V(G) \rightarrow \mathbb Z_+$ such that $$d(u,v)+|c(u)-c(v)|\geq 1+\text{diam}(G)$$ for every two distinct vertices $u$ and $v$ of $G$ (where $d(u,v)$ is the distance between $u$ and $v$). The span of a radio labeling is the maximum integer assigned to a vertex. The radio number of a graph $G$ is the mi…
Let $G$ be a finite, connected, undirected graph with diameter $diam(G)$ and $d(u,v)$ denote the distance between $u$ and $v$ in $G$. A radio labeling of a graph $G$ is a mapping $f: V(G) \rightarrow \{0,1,2,...\}$ such that $|f(u)-f(v)| \geq diam(G) + 1 - d(u,v)$ for every pair of distinct vertices $u, v$ of $G$. The radio number of $G$, denoted by $rn(G)$, is the smallest integer $k$ such that $G$ has a radio labeling $f$ with $\max\{f(v) : v \in V(G)\} = k$. In this paper, we determine the radio number for three families of trees obtained by taking graph operation on a given tree or a famil…
A graceful labeling of a graph $G$ with $m$ edges consists of labeling the vertices of $G$ with distinct integers from $0$ to $m$ such that, when each edge is assigned as induced label the absolute difference of the labels of its endpoints, all induced edge labels are distinct. Rosa established two well known conjectures: all trees are graceful (1966) and all triangular cacti are graceful (1988). In order to contribute to both conjectures we study graceful labelings in the context of graph games. The Graceful game was introduced by Tuza in 2017 as a two-players game on a connected graph in whi…
Let $\mathbb{N}$ be the set of positive integers. A radio labeling of a graph $G$ is a mapping $φ: V(G) \rightarrow \mathbb{N} \cup \{0\}$ such that the inequality $|φ(u)-φ(v)| \geq diam(G) + 1 - d(u,v)$ holds for every pair of distinct vertices $u,v$ of $G$, where $diam(G)$ and $d(u,v)$ are the diameter of $G$ and distance between $u$ and $v$ in $G$, respectively. The radio number $rn(G)$ of $G$ is the smallest number $k$ such that $G$ has radio labeling $φ$ with $\max\{φ(v) : v \in V(G)\}$ = $k$. Das et al. [Discrete Math. $\mathbf{340}$(2017) 855-861] gave a technique to find a lower bound …
A radio labeling of a graph $G$ is a mapping $f$ : $V(G) \rightarrow \{0, 1, 2,...\}$ such that $|f(u)-f(v)| \geq diam(G) + 1 - d(u,v)$ holds for every pair of vertices $u$ and $v$, where $diam(G)$ is the diameter of $G$ and $d(u,v)$ is the distance between $u$ and $v$ in $G$. The radio number of $G$, denoted by $rn(G)$, is the smallest $t$ such that $G$ admits a radio labeling with $t=\max\{|f(v)-f(u)|: v, u \in V(G)\}$. A block graph is a graph such that each block (induced maximal 2-connected subgraph) is a complete graph. In this paper, a lower bound for the radio number of block graphs is…
For a simple finite connected graph $G$, let $diam(G)$ and $d_{G}(u,v)$ denote the diameter of $G$ and distance between $u$ and $v$ in $G$, respectively. A radio labeling of a graph $G$ is a mapping $f$ : $V(G) \rightarrow \{0, 1, 2,...\}$ such that $|f(u)-f(v)| \geq diam(G) + 1 - d_{G}(u,v)$ holds for every pair of distinct vertices $u,v$ of $G$. The radio number $rn(G)$ of $G$ is the smallest number $k$ such that $G$ has radio labeling $f$ with max$\{f(v):v \in V(G)\}$ = $k$. Bantva et al. gave a lower bound for the radio number of trees in [Lemma 3.1, Discrete Applied Math.,217(2017),110-12…
