Abstract
Given a simple graph G, the vertex partition Π: V(G)=V1∪V2∪⋯∪Vr is said to be an equitable partition if, for any u∈Vi, |Vj∩NG(u)|=bij is a constant whenever 1≤i,j≤r. An equitable partition Π leads to a divisor G/Π of G, which is the directed multigraph with vertices V1,V2,…,Vr and bij arcs from Vi to Vj. Conversely, a directed multigraph may not be a divisor of some simple graph. In this paper we give a necessary and sufficient condition for a directed multigraph to be the divisor of some simple graph. By the way, we give a method to construct many classes of connected graphs with exactly k main eigenvalues for any positive integer k.
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