The number of rooted forests in circulant graphs

The Zero Forcing Number of Circulant Graphs

The zero forcing number of a graph G is the cardinality of the smallest subset of the vertices of G that forces the entire graph using a color change rule. This paper presents some basic properties of circulant graphs and later investigates zero forcing numbers of circulant graphs of the form C[n, {s, t}], while also giving attention to propagation time for specific zero forcing sets.

The number of spanning trees in circulant graphs

The Asymptotic Number of Spanning Trees in Circulant Graphs

Let T (G) be the number of spanning trees in graph G. In this note we explore the asymptotics of T (G) for circulant graphs. The circulant graph Cs1,s2,···,sk n is the 2k regular graph with n vertices labelled 0, 1, 2, · · · , n − 1, where node i has the 2k neighbors, (0 ≤ i ≤ n − 1) adjacent to vertices i + s1, i + s2, · · · , i + sk mod n. In this note we give a closed formula for the asympto...

Counting rooted spanning forests in complete multipartite graphs

Jin and Liu discovered an elegant formula for the number of rooted spanning forests in the complete bipartite graph a1;a2 , with b1 roots in the rst vertex class and b2 roots in the second vertex class. We give a simple proof to their formula, and a generalization for complete m-partite graphs, using the multivariate Lagrange inverse. Y. Jin and C. Liu [3] give a formula for f(m; l;n; k), the n...

On the Lovász Number of Certain Circulant Graphs

The theta function of a graph, also known as the Lovv asz number, has the remarkable property of being computable in polynomial time, despite being \sandwiched" between two hard to compute integers, i.e., clique and chromatic number. Very little is known about the explicit value of the theta function for special classes of graphs. In this paper we provide the explicit formula for the Lovv asz n...