Abstract
This article introduces the concept of a D-levels locally homogeneous graph which forms a special type of self-centered graphs. Some results relating D-levels locally homogeneous graphs to diametrical graphs (which are also self-centered) are obtained. Diametrical graphs of small order (≤ 8) are shown to be D-levels locally homogeneous. When the diameter is 2, it is shown that the concept of a D-levels locally homogeneous graph and that of a diametrical graph coincide. Bipartite diametrical graphs of diameter 3 and regular S-graphs of diameter 4 are characterized as D-levels locally homogeneous graphs. We also prove that the vertex-transitive graphs are precisely the totally D-levels locally homogeneous graphs.
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