Regular Triangulated Toroidal Graphs with Applications to Cellular and Interconnection Networks

This paper introduces a special graph family, together with an extensive characterization of some of its properties and two of its immediate applications. The graph denoted by Ti,j, is a regular hexavalent toroidal graph. The topological features of Ti,j include vertex symmetry, Hamiltonian decomposition, translation and rotation isomorphism. Topologically, the graph is a torus, while algebraically, it can also be expressed as a Cayley graph, defined on the cyclic group <1, k, k, −1, −k, − k>, where k can be determined from the (i,j) parameters defining graph Ti,j. As a direct consequence, the proposed graph, which has j i j i j j i i j i ⋅ − + = + ⋅ + = 2 2 2 , ) ( ρ vertices, is vertex-symmetric. For the special case when i − j is a multiple of 3, the graph has a unique 3-coloring. The diameter of the graph can also be expressed as a function of i and j: ( )   3 / 2 j i d + = . As a result of its highly symmetric topology, the graph is employed in modeling and analysis of cellular and interconnection networks. A more appropriate way of modeling highly dense cellular networks is shown to be the model using the triangular lattice in which the nodes represent the transceivers of the network, rather than the traditional hexagonal lattice where coverage overlap regions cannot be explicitly represented. The toroidal embedding of the triangular lattice using the Ti,j, graph helps us model and simulate the functionality of a cellular network with strong overlap regions without inducing any artifacts due to boundary effects limitations, simultaneously preserving the regularity of the graph model. For the constructing parameters of Ti,j, such that j = i − 1, the graph is optimal with maximum connectivity and maximum number of vertices for a locally planar graph, given diameter d, which in this case will be equal to j. This feature makes the Ti,j graph desirable for interconnection networks topology, together with the graph vertices’ algorithmic labeling scheme, presented in this paper. Communicated by Balaji Raghavachari: submitted May 1999; revised May 2001 and August 2002. Iridon and Matula, A Torus Graph Family. JGAA 6(4) 373-404 (2002) 374