We use throughout the notation of [5], to which we refer for the basic theory of finite rank 3 permutation groups G. The solvable primitive rank 3 permutation groups have been determined by Foulser [4] and, independently, by Dornhoff [2]. Since rank 3 groups of odd order are solvable we assume that G has even order. Then the graphs JA and JT associated with the non-trivial orbitals A and T of G...

Graphs are everywhere: social networks, the World Wide Web, biological networks, and many more. The sizes of graphs are growing at unprecedented rate, spanning millions and billions of nodes and edges. What are the patterns in large graphs, spanning Giga, Tera, and heading toward Peta bytes? What are the best tools, and how can they help us solve graph mining problems? How do we scale up algori...

Vertices x and y are a dominant pair in a tournament T if for all vertices z 6 = x; y either x beats z or y beats z. Vertices x and y are a compliant pair in a tournament T if for all vertices z 6 = x; y either z beats x or z beats y. Let DC (T) be the graph on the same vertex set as T with edges between pairs of vertices that are either a dominant pair or a compliant pair in T. We show that th...

Many tasks in NLP stand to benefit from robust measures of semantic similarity for units above the level of individual words. Rich semantic resources such as WordNet provide local semantic information at the lexical level. However, effectively combining this information to compute scores for phrases or sentences is an open problem. Our algorithm aggregates local relatedness information via a ra...

We propose the use of the Gabriel graph for the exploratory analysis of potentially high dimensional labeled data. Gabriel graph is a subgraph of the Delaunay triangulation, which connects two data points vi and vj for which there is no other point vk inside the open ball with diameter [vivj ]. If all the Gabriel neighbors of a datum have a different class than its own, this datum is said to be...

A pair of complementary algorithms are presented. One of the pair is a fast method for connecting graphs with an edge. The other is a fast method for removing edges from a graph. Both algorithms employ the same tree-based graph representation and so, in concert, can arbitrarily modify any graph. Since the clusters of a percolation model may be described as simple connected graphs, an efficient ...

A connected graph G is said to be neighbourly irregular graph if no two adjacent vertices of G have same degree. In this paper we obtain neighbourly irregular derived graphs such as semitotal-point graph, k^{tℎ} semitotal-point graph, semitotal-line graph, paraline graph, quasi-total graph and quasivertex-total graph and also neighbourly irregular of some graph products.

the energy of a graph is equal to the sum of the absolute values of its eigenvalues. two graphs of the same order are said to be equienergetic if their energies are equal. we point out the following two open problems for equienergetic graphs. (1) although it is known that there are numerous pairs of equienergetic, non-cospectral trees, it is not known how to systematically construct any such pa...

let $r$ be a ring with unity. the undirected nilpotent graph of $r$, denoted by $gamma_n(r)$, is a graph with vertex set ~$z_n(r)^* = {0neq x in r | xy in n(r) for some y in r^*}$, and two distinct vertices $x$ and $y$ are adjacent if and only if $xy in n(r)$, or equivalently, $yx in n(r)$, where $n(r)$ denoted the nilpotent elements of $r$. recently, it has been proved that if $r$ is a left ar...