A graph G= (V, E) is called a hyper-ring with N nodes (N-HR for short) if V = {0, ..., N-1} and E = {(u,v)| v-u modulo N is a power of 2}. In other words, a hyper-ring with N nodes is a graph in which all of the edges connect nodes that are at distances of powers of 2 from each other. Hyper-rings are a multi-machine organization that is, in a sense, a generalization of the standard hyper-cube a...

Bubble-sort, macro-star, and transposition graphs are interconnection networks with the advantages of star graphs in terms of improving the network cost of a hypercube. These graphs contain a star graph as their sub-graph, and have node symmetry, maximum fault tolerance, and recursive partition properties. This study proposes embedding methods for these graphs based on graph definitions, and sh...

The unitary Cayley graph Xn has vertex set Zn = {0, 1,…, n-1} and vertices u and v are adjacent, if gcd(uv, n) = 1. In [A. Ilić, The energy of unitary Cayley graphs, Linear Algebra Appl. 431 (2009) 1881–1889], the energy of unitary Cayley graphs is computed. In this paper the Wiener and hyperWiener index of Xn is computed.

Let $G$ be a graph. Let $f:V(G)to{0,1,2, ldots, k-1}$ be a map where $k in mathbb{N}$ and $k>1$. For each edge $uv$, assign the label $left|f(u)-f(v)right|$. $f$ is called a $k$-total difference cordial labeling of $G$ if $left|t_{df}(i)-t_{df}(j)right|leq 1$, $i,j in {0,1,2, ldots, k-1}$ where $t_{df}(x)$ denotes the total number of vertices and the edges labeled with $x$.A graph with admits a...

The (n, k)-star graph is a generalized version of the n-star graph, which belongs to the class of Cayley graphs, and has been recognized as an attractive alternative to an n-cube for building massively parallel computers. Recently, Chen et al. showed that an (n, k)-star graph is 6-weak-vertex-pancyclic for k < n–1, that is, each vertex of an (n, k)-star graph is contained in a cycle of length r...

Hypercubes and star graphs are widespread topologies of interconnection networks. The class of hyper-stars was introduced as a new type of interconnection network to compete with both hypercubes and star graphs, and the class of folded hyper-stars is a strengthened variation of hyperstars with additional links to connect nodes with complemented 0/1-strings. Constructing independent spanning tre...

Many theoretical-based comparison studies, relying on graph structural and algorithmic properties, have been conducted for the hypercube and the star graph. None of these studies, however, have considered real working conditions and implementation limits. We have compared the performance of the star and hypercube networks for different message lengths and number of virtual channels, and conside...

Akers et al. (Proceedings of the International Conference on Parallel Processing, 1987, pp. 393–400) proposed an interconnection topology, the star graph, as an alternative to the popular n-cube. Jwo et al. (Networks 23 (1993) 315–326) studied the alternating group graph An. Cheng et al. (Super connectivity of star graphs, alternating group graphs and split-stars, Ars Combin. 59 (2001) 107–116)...

A star coloring of a graph is a proper coloring such that no path on four vertices is 2-colored. We prove that every planar graph with girth at least 9 can be star colored using 5 colors, and that every planar graph with girth at least 14 can be star colored using 4 colors; the figure 4 is best possible. We give an example of a girth 7 planar graph that requires 5 colors to star color.

the unitary cayley graph xn has vertex set zn = {0, 1,…, n-1} and vertices u and v areadjacent, if gcd(uv, n) = 1. in [a. ilić, the energy of unitary cayley graphs, linear algebraappl. 431 (2009) 1881–1889], the energy of unitary cayley graphs is computed. in this paperthe wiener and hyperwiener index of xn is computed.