Network reliability analysis by counting the number of spanning trees

Abstract

In this paper, we consider problems related to the network reliability problem restricted to circulant graphs (networks). Let 1/spl les/s/sub 1/<s/sub 2/<...<s/sub k//spl les/[n/2] be given integers. An undirected circulant graph, C/sub n//sup s1,s2,...,sk/, has n vertices 0, 1, 2, ..., n-1, and for each s/sub i/ (1/spl les/i/spl les/k) and j (0/spl les/j/spl les/n-1) there is an edge between j and j+s/sub i/ mod n. Let T(C/sub n//sup s1,s2,...,sk/) stand for the number of spanning trees of C/sub n//sup s1,s2,...,sk/. For this special class of graphs, a general and most recent result is obtained by Y. P. Zhang et al (Discrete Mathematics vol. 223, pp.337-350, 2000) where it is shown that T(C/sub n//sup s1,s2,...,sk/)=na/sub n//sup 2/ where a/sub n/ satisfies a linear recurrence relation of order 2/sup sk-1/. In this paper we obtain further properties of the numbers a/sub n/ by considering their combinatorial structures. Using these properties we investigate the open problem posed in the Conclusion of Y. P. Zhang et al. We describe our technique and asymptotic properties of the numbers, using examples.