@MISC{05boundaryquotient, author = {}, title = {BOUNDARY QUOTIENT GRAPHS AND THE GRAPH INDEX}, year = {2005} }

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Abstract

Abstract. In this paper, we will consider the boundary quotient graphs. Let G be a finite directed graph with its vertex set V (G) and its edge set E(G). The boundary ∂ of G is a subset of the vertex set V (G). For the given boundary ∂ ⊆ V (G), we give an boundary quotient: if v1, v2 ∈ ∂, then v1 = v2, for all v1, v2 ∈ ∂ Then we can construct a new graph G ∂ = G / ∂ called the ∂-quotient graph of G. In Chapter 1, we restrict our interests to the finite simplicial directed graphs. We will observe some properties of G / ∂. In particular, we show that all total boundary quotient graphs has the same type, where total boundary ∂ is V (G). Every total boundary quotient graph is graph-isomorphic to onevertex-|E(G)|-loop-edge graph. In fact, every total boundary quotient graph of a finite directed graph is graph-isomorphic to the one-vertex-multi-loopedge graph. This result shows that boundary quotient ∂ is not an invariants on finite simplicial directed graphs. However, we show that the “admissible” boundary quotient is an invariant on finite simplicial directed graphs with mixed maximal types. In Chapter 2, we consider arbitrary finite directed