We show that recursive circulant G(cd m ; d) is hamiltonian decomposable. Recursive circulant is a graph proposed for an interconnection structure of multicomputer networks in [8]. The result is not only a partial answer to the problem posed by Alspach that every connected Cayley graph over an abelian group is hamiltonian decomposable, but also an extension of Micheneau's that recursive circula...

In this paper, we study matching integral graphs of small order. A graph is called matching integral if the zeros of its matching polynomial are all integers. Matching integral graphs were first studied by Akbari, Khalashi, etc. They characterized all traceable graphs which are matching integral. They studied matching integral regular graphs. Furthermore, it has been shown that there is no matc...

In this paper we study defensive alliances in some regular graphs. We determine which subgraphs could a critical defensive alliance of a graph G induce, if G is 6-regular and the cardinality of the alliance is at most 8. In particular, we study the case of circulant graphs, i.e. Cayley graphs on a cyclic group. The critical defensive alliances of a circulant graph of degree at most 6 are comple...

We present novel families of wavelets and associated filterbanks for the analysis and representation of functions defined on circulant graphs. In this work, we leverage the inherent vanishing moment property of the circulant graph Laplacian operator, and by extension, the e-graph Laplacian, which is established as a parameterization of the former with respect to the degree per node, for the des...

Semi-directed 2-circulant graph is a subgraph of an (undirected) 2-circulant graph in which the links of one type (i.e., short or long) are directed while the other links are undirected. The shortest paths in semi-directed circulant graphs are called the restricted shortest paths in 2-circulant graphs. In this paper we show that the problem of finding the restricted shortest paths is equivalent...

It has long been known that the number of spanning trees in circulant graphs with fixed jumps and n nodes satisfies a recurrence relation in n. The proof of this fact was algebraic (relating the products of eigenvalues of the graphs’ adjacency matrices) and not combinatorial. In this paper we derive a straightforward combinatorial proof of this fact. Instead of trying to decompose a large circu...

A circulant (di)graph is a (di)graph on n vertices that admits a cyclic automorphism of order n. This paper provides a survey of the work that has been done on finding the automorphism groups of circulant (di)graphs, including the generalisation in which the edges of the (di)graph have been assigned colours that are invariant under the aforementioned cyclic automorphism. Mathematics Subject Cla...

A circulant is a Cayley graph of a cyclic group Arc transitive circulants of square free order are classi ed It is shown that an arc transitive circulant of square free order n is one of the following the lexicographic product Kb or the deleted lexicographic Kb b where n bm and is an arc transitive circulant or is a normal circulant that is Aut has a normal regular cyclic subgroup

Kirchhoo's Matrix Tree Theorem permits the calculation of the number of spanning trees in any given graph G through the evaluation of the determinant of an associated matrix. In the case of some special graphs Boesch and Prodinger 9] have shown how to use properties of Chebyshev polyno-mials to evaluate the associated determinants and derive closed formulas for the number of spanning trees of g...