The incidence algebraI(X,ℝ)of a locally finite poset(X,≤)has been defined and studied by Spiegel and O'Donnell (1997). A poset(V,≤)has a directed graph(Gv,≤)representing it. Conversely, any directed graphGwithout any cycle, multiple edges, and loops is represented by a partially ordered setVG. So in this paper, we define an incidence algebraI(G,ℤ)for(G,≤)overℤ, the ring of integers, byI(G,ℤ)={fi,fi*:V×V→ℤ}wherefi(u,v)denotes the number of directed paths of lengthifromutovandfi*(u,v)=−fi(u,v). WhenGis finite of ordern,I(G,ℤ)is isomorphic to a subring ofMn(ℤ). Principal idealsIvof(V,≤)induce the subdigraphs〈Iv〉which are the principal idealsℐvof(Gv,≤). They generate the idealsI(ℐv,ℤ)ofI(G,ℤ). These results are extended to the incidence algebra of the digraph representing a locally finite weak poset both bounded and unbounded.

Let S k+1 denote a star with k edges. Tarsi and Yamamoto et al. have characterized the S k+1 -decomposability of K n the complete graph. In this paper we study the edge decomposition of both K n and the complete bipartite graph K m,n into copies of the union of two edge disjoint stars S p+1 and S q+1 where p ≠q and p,q 2 and obtain the necessary and sufficient conditions for the S p+1 ∪S q+1 -decomposability of K n and K m,n .