On the independent spanning trees of recursive circulant graphs [formula omitted] with [formula omitted

Abstract

Two spanning trees of a graph G are said to be independent if they are rooted at the same vertex r , and for each vertex v ≠ r in G , the two different paths from v to r , one path in each tree, are internally disjoint. A set of spanning trees of G is independent if they are pairwise independent. The construction of multiple independent spanning trees has many applications in network communication. For instance, it is useful for fault-tolerant broadcasting and secure message distribution. A recursive circulant graph G ( N , d ) has N = c d m vertices labeled from 0 to N − 1 , where d ⩾ 2 , m ⩾ 1 , and 1 ⩽ c < d , and two vertices x , y ∈ G ( N , d ) are adjacent if and only if there is an integer k with 0 ⩽ k ⩽ ⌈ log d N ⌉ − 1 such that x ± d k ≡ y (mod N ). In this paper, we propose an algorithm to construct multiple independent spanning trees on a recursive circulant graph G ( c d m , d ) with d > 2 , where the number of independent spanning trees matches the connectivity of G ( c d m , d ) .