We consider unitary graphs attached to d n , for odd integer n, using an analogue of the Euclidean distance. These graphs are shown to be integral.

We consider the maximum integral tWo-Commodity flow problem in augmented planar graphs (i.e., with both source-sink edges added) and provide an o(IVI 2 1ogIVI) algorithm for that problem. Let G = (V, E) be a graph and w : E-t Z+ a weight function. Let T C V be an even subset of the vertices of G. A T-cut is an edge-cutset of the graph which divides T into two odd sets. Lomonosov gave a good cha...

Integral sets of finite groups are discussed and related to the integral Cayley graphs. The Boolean algebra of integral sets are determined for dihedral group and finite abelian groups. We characterize the finite abelian groups as those finite groups where the Boolean algebra generated by integral sets equals the Boolean algebra generated by its subgroups.

This paper investigates consensus problems for continuous-time multiagent systems with time-varying communication graphs subject to input noise. Based on input-to-state stability and integral input-to-state stability, robust consensus and integral robust consensus are defined with respect to L∞ and L1 norms of the noise function, respectively. Sufficient and/or necessary connectivity conditions...

In this white paper we report the work that was done on the problem of generation combinatorial structures with some rare invariant properties. These combinatorial structures are connected integral graphs. All (588) of such graphs of order 1 ≤ n ≤ 12 are known. The main goal of this work was to reduce the time of generation by distributing graph generators over hosts in PRACE-RI, and to reduce ...

We show if the flow polynomial of a bridgeless graph G has only integral roots, then G is the dual graph to a planar chordal graph. We also show that for 3-connected cubic graphs, the same conclusion holds under the weaker hypothesis that it has only real flow roots. Expressed in the language of matroid theory, this result says that the cographic matroids with only integral characteristic roots...

The existence of perfect state transfer in quantum spin networks based on integral circulant graphs has been considered recently by Saxena, Severini and Shparlinski. We give the simple condition for characterizing integral circulant graphs allowing the perfect state transfer in terms of its eigenvalues. Using that we complete the proof of results stated by Saxena, Severini and Shparlinski. More...

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We show if the flow polynomial of a bridgeless graph G has only integral roots, then G is the dual graph to a planar chordal graph. We also show that for 3-connected cubic graphs, the same conclusion holds under the weaker hypothesis that it has only real flow roots. Expressed in the language of matroid theory, this result says that the cographic matroids with only integral characteristic roots...

The intention of the paper is to move a step towards a classification of network topologies that exhibit periodic quantum dynamics. We show that the evolution of a quantum system whose hamiltonian is identical to the adjacency matrix of a circulant graph is periodic if and only if all eigenvalues of the graph are integers (that is, the graph is integral). Motivated by this observation, we focus...