Search results for "Vertex-transitive graph"
showing 6 items of 6 documents
Incomplete vertices in the prime graph on conjugacy class sizes of finite groups
2013
Abstract Given a finite group G, consider the prime graph built on the set of conjugacy class sizes of G. Denoting by π 0 the set of vertices of this graph that are not adjacent to at least one other vertex, we show that the Hall π 0 -subgroups of G (which do exist) are metabelian.
Optical Routing of Uniform Instances in Cayley Graphs
2001
Abstract Abstract We consider the problem of routing uniform communication instances in Cayley graphs. Such instances consist of all pairs of nodes whose distance is included in a specified set U. We give bounds on the load induced by these instances on the links and for the wavelength assignment problem as well. For some classes of Cayley graphs that have special symmetry property (rotational graphs), we are able to construct routings for uniform instances such that the load is the same for each link of the graph.
A comparison of compatible, finite, and inductive graph properties
1993
Abstract In the theory of hyperedge-replacement grammars and languages, one encounters three types of graph properties that play an important role in proving decidability and structural results. The three types are called compatible, finite, and inductive graph properties. All three of them cover graph properties that are well-behaved with respect to certain operations on hypergraphs. In this paper, we show that the three notions are essentially equivalent. Consequently, three lines of investigation in the theory of hyperedge replacement - so far separated - merge into one.
P-matrix completions under weak symmetry assumptions
2000
An n-by-n matrix is called a Π-matrix if it is one of (weakly) sign-symmetric, positive, nonnegative P-matrix, (weakly) sign-symmetric, positive, nonnegative P0,1-matrix, or Fischer, or Koteljanskii matrix. In this paper, we are interested in Π-matrix completion problems, that is, when a partial Π-matrix has a Π-matrix completion. Here, we prove that a combinatorially symmetric partial positive P-matrix has a positive P-matrix completion if the graph of its specified entries is an n-cycle. In general, a combinatorially symmetric partial Π-matrix has a Π-matrix completion if the graph of its specified entries is a 1-chordal graph. This condition is also necessary for (weakly) sign-symmetric …
A dual of 4-regular graph forG × C2n
2003
Abstract A graph is said h-decomposable if its edge-set is decomposable into edge-disjoint hamiltonian cycles. Jha [3] conjectured that if G is a non-bipartite h-decomposable graph on even number of vertices, then G × K2 is h-decomposable. We use the notion of dual graph defined in [4], we prove that if G = Q1,2 ⊕ C3,4 is a 4-regular non-bipartite h-decomposable graph and the dual graphs relative to Q1,2 and C3,4 are connected then G × K 2 and G × C 2n are h-decomposable (where C 2n is an even cycle).
Generalizations of the periodicity Theorem of Fine and Wilf
2005
We provide three generalizations to the two-dimensional case of the well known periodicity theorem by Fine and Wilf [4] for strings (the one-dimensional case). The first and the second generalizations can be further extended to hold in the more general setting of Cayley graphs of groups. Weak forms of two of our results have been developed for the design of efficient algorithms for two-dimensional pattern matching [2, 3, 6].