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Quotient-polynomial graphs. (English) Zbl 1326.05174
Summary: As a generalization of orbit-polynomial and distance-regular graphs, we introduce the concept of a quotient-polynomial graph. In these graphs every vertex $$u$$ induces the same regular partition around $$u$$, where all vertices of each cell are equidistant from $$u$$. Some properties and characterizations of such graphs are studied. For instance, all quotient-polynomial graphs are walk-regular and distance-polynomial. Also, we show that every quotient-polynomial graph generates a (symmetric) association scheme.

##### MSC:
 05E30 Association schemes, strongly regular graphs 05E05 Symmetric functions and generalizations 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 05C12 Distance in graphs 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
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