For a symmetric cellular algebra, we study properties of the dual basis of a cellular basis first. Then a nilpotent ideal is constructed. The ideal connects the radicals of cell modules with the radical of the algebra. It also yields some information on the dimensions of simple modules. As a by-product, we obtain some equivalent conditions for a finite-dimensional symmetric cellular algebra to be semisimple.

We show that the out-radius and the radius grow linearly, or "almost" linearly, in iterated line digraphs. Further, iterated line digraphs with a prescribed out-center, or a center, are constructed. It is shown that not every line digraph is admissible as an out-center of line digraph.

A radio antipodal coloring of a connected graph $G$ with diameter $d$ is an assignment of positive integers to the vertices of $G$, with $x\in V\left(G\right)$ assigned $c\left(x\right)$, such that $$d(u,v)+\left|c\right(u)-c(v\left)\right|\ge d$$ for every two distinct vertices $u$, $v$ of $G$, where $d(u,v)$ is the distance between $u$ and $v$ in $G$. The radio antipodal coloring number $\mathrm{a}c\left(c\right)$ of a radio antipodal coloring $c$ of $G$ is the maximum color assigned to a vertex of $G$. The radio antipodal chromatic number $\mathrm{a}c\left(G\right)$ of $G$ is $min\left\{\mathrm{a}c\right(c\left)\right\}$ over all radio antipodal colorings $c$ of $G$. Radio antipodal chromatic numbers of paths...

For k ∈ ℤ+ and G a simple, connected graph, a k-radio labeling f : V (G) → ℤ+ of G requires all pairs of distinct vertices u and v to satisfy |f(u) − f(v)| ≥ k + 1 − d(u, v). We consider k-radio labelings of G when k = diam(G). In this setting, f is injective; if f is also surjective onto {1, 2, . . . , |V (G)|}, then f is a consecutive radio labeling. Graphs that can be labeled with such a labeling are called radio graceful. In this paper, we give two results on the existence of radio graceful...

For a connected graph G of diameter d and an integer k with 1 ≤ k ≤ d, a radio k-coloring of G is an assignment c of colors (positive integers) to the vertices of G such that d(u,v) + |c(u)- c(v)| ≥ 1 + k for every two distinct vertices u and v of G, where d(u,v) is the distance between u and v. The value rcₖ(c) of a radio k-coloring c of G is the maximum color assigned to a vertex of G. The radio k-chromatic number rcₖ(G) of G is the minimum value of rcₖ(c) taken over all radio k-colorings c of...

Frequency planning consists in allocating frequencies to the transmitters of a cellular network so as to ensure that no pair of transmitters interfere. We study the problem of reducing interference by modeling this by a radio k-labeling problem on graphs: For a graph G and an integer k ≥ 1, a radio k-labeling of G is an assignment f of non negative integers to the vertices of G such that $|f\left(x\right)-f\left(y\right)|\ge k+1-d_G(x,y)$, for any two vertices x and y, where $d_G(x,y)$ is the distance between x and y in G. The radio k-chromatic number is...

For a graph G and any two vertices u and v in G, let d(u,v) denote the distance between u and v and let diam(G) be the diameter of G. A multilevel distance labeling (or radio labeling) for G is a function f that assigns to each vertex of G a positive integer such that for any two distinct vertices u and v, d(u,v) + |f(u) - f(v)| ≥ diam(G) + 1. The largest integer in the range of f is called the span of f and is denoted span(f). The radio number of G, denoted rn(G), is the minimum span of any radio...

A radio labeling is an assignment c:V(G) → N such that every distinct pair of vertices u,v satisfies the inequality d(u,v) + |c(u)-c(v)| ≥ 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 ≥ 1, n ≥ s, have vertex set (i,j) | i = 1,2 and j = 1,...,n and edge set ((i,j),(i,j ±1)) ∪ ((1,i),(2,i+σ)) | σ = -⌊(s-1)/2⌋...,0,...,⌊s/2⌋. In this paper we determine the radio...

The eccentricity $e\left(v\right)$ of a vertex $v$ is defined as the distance to a farthest vertex from $v$. The radius of a graph $G$ is defined as a $r\left(G\right)={min}_{u\in V\left(G\right)}\left\{e\left(u\right)\right\}$. A graph $G$ is radius-edge-invariant if $r(G-e)=r\left(G\right)$ for every $e\in E\left(G\right)$, radius-vertex-invariant if $r(G-v)=r\left(G\right)$ for every $v\in V\left(G\right)$ and radius-adding-invariant if $r(G+e)=r\left(G\right)$ for every $e\in E\left(\overline{G}\right)$. Such classes of graphs are studied in this paper.

Let $G$ be a nontrivial connected graph on which is defined a coloring $cE\left(G\right)\to \{1,2,...,k\}$, $k\in \mathbb{N}$, of the edges of $G$, where adjacent edges may be colored the same. A path $P$ in $G$ is a rainbow path if no two edges of $P$ are colored the same. The graph $G$ is rainbow-connected if $G$ contains a rainbow $u-v$ path for every two vertices $u$ and $v$ of $G$. The minimum $k$ for which there exists such a $k$-edge coloring is the rainbow connection number $\mathrm{r}c\left(G\right)$ of $G$. If for every pair $u,v$ of distinct vertices, $G$ contains a rainbow $u-v$ geodesic, then $G$ is...

An edge-coloured connected graph G = (V,E) is called rainbow-connected if each pair of distinct vertices of G is connected by a path whose edges have distinct colours. The rainbow connection number of G, denoted by rc(G), is the minimum number of colours such that G is rainbow-connected. In this paper we prove that rc(G) ≤ k if |V (G)| = n and for all integers n and k with n − 6 ≤ k ≤ n − 3. We also show that this bound is tight.

An edge-colored graph G is rainbow connected, if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph G, denoted rc(G), is the smallest number of colors that are needed in order to make G rainbow connected. In this paper we show that rc(G) ≤ 3 if |E(G)| ≥ [...] + 2, and rc(G) ≤ 4 if |E(G)| ≥ [...] + 3. These bounds are sharp.

A path in an edge-colored graph G is rainbow if no two edges of the path are colored the same. The rainbow connection number rc(G) of G is the smallest integer k for which there exists a k-edge-coloring of G such that every pair of distinct vertices of G is connected by a rainbow path. Let f(d) denote the minimum number such that rc(G) ≤ f(d) for each bridgeless graph G with diameter d. In this paper, we shall show that 7 ≤ f(3) ≤ 9.