We investigate subsets of the symmetric group with structure similar to that of a graph. The “trees” of these subsets then lead to minimal highly symmetric generating sets of the symmetric group. We show that there exist generating sets among these with edge-transitive Cayley graphs and investigate them in relation to the Lovasz conjecture.

The 2-restricted edge-connectivity λ′′ of a graph G is defined to be the minimum cardinality |S| of a set S of edges such thatG−S is disconnected and is of minimum degree at least two. It is known that λ′′ ≤ g(k − 2) for any connected k-regular graph G of girth g other than K4, K5 and K3,3, where k ≥ 3. In this paper, we prove the following result: For a connected vertex-transitive graph of ord...

A module of an undirected graph is a set X of nodes such for each node x not in X , either every member of X is adjacent to x, or no member of X is adjacent to x. There is a canonical linear-space representation for the modules of a graph, called the modular decomposition. Closely related to modular decomposition is the transitive orientation problem, which is the problem of assigning a directi...

Knödel graphs and Fibonacci graphs are two classes of bipartite incident-graph of circulant digraphs. Both graphs have been extensively studied for the purpose of fast communications in networks, and they have deserved a lot of attention in this context. In this paper, we show that there exists an O(n log n)-time algorithm to recognize Knödel graphs, and that the same technique applies to Fibon...

The resultant of a square system of homogeneous polynomials is a polynomial in their coefficients which vanishes whenever the system has a solution. Canny gave an algorithm running in polynomial space to compute it but no lower bound was known. We investigate the complexity of the associated decision problem and give a hardness result: Testing the resultant for zero lies in the class Arthur −Me...

We study the problem of maintaining recursively-de ned views, such as the transitive closure of a relation, in traditional relational languages that do not have recursion mechanisms. In particular, we show that the transitive closure cannot be maintained in relational calculus under deletion of edges. We use new proof techniques to show this result. These proof techniques generalize to other la...

A graph is half-arc-transitive if its automorphism group acts transitively on its vertex set and edge set, but not arc set. It was shown by [Y.-Q. Feng, K.S. Wang, C.X. Zhou, Tetravalent halftransitive graphs of order 4p, European J. Combin. 28 (2007) 726–733] that all tetravalent half-arc-transitive graphs of order 4p for a prime p are non-Cayley and such graphs exist if andonly if p−1 is divi...

We describe the generation of all possible shell and dome shapes that can be uniquely meshed (tiled) using a single type of mesh face (tile), and following a single meshing (tiling) rule that governs the mesh (tile) arrangement with maximal vertex, edge and face symmetries. Such tiling arrangements or congruently tiled meshed shapes, are frequently found in chemical forms (fullerenes or Bucky b...

A subgroup of the automorphism group a graph Γ is said to be half-arc-transitive on if its action transitive vertex set and edge but not arc Γ. Tetravalent graphs girths 3 4 admitting automorphisms have previously been characterized. In this paper we study examples girth 5. We show that, with two exceptions, all such only directed 5-cycles respect corresponding induced orientation edges. Moreov...

In distributed networks, a target party T could be a person never meet with a source party S, therefore S may not hold any prior evaluation of trustworthiness of T . To get permit to access S, T should be somewhat trusted by S. Consequently, we should study the approach to evaluate trustworthiness of T . To attack the problem, we view individual participant in distributed networks as a node of ...