Partially Distance-regular Graphs and Partially Walk-regular Graphs∗
نویسندگان
چکیده
We study partially distance-regular graphs and partially walkregular graphs as generalizations of distance-regular graphs and walkregular graphs respectively. We conclude that the partially distanceregular graphs can be viewed as some extremal graphs of partially walk-regular graphs. In the special case that the graph is assumed to be regular with four distinct eigenvalues, a well known class of walkregular graphs, we show that there exists a rational function f in the expression of the order and the four eigenvalues of the graph such that k2(x), the number of vertices with distance 2 to a vertex x, satisfies k2(x) ≥ f ; furthermore we show the equality holds for each vertex x if and only if the graph is distance-regular with diameter 3.
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