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On Graphs Defined by Some Systems of Equations

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In this talk I will present a simple method for constructing infinite families of graphs defined by a class of systems of equations over commutative rings. The graphs in all such families possess some general properties including regularity or bi-regularity, existence of special vertex colorings, and existence of covering maps between every two members of the same family (hence, embedded spectra). Another general property is that nearly every graph constructed in this manner edge-decomposes either the complete, or complete bipartite, graph which it spans.

In many instances, specializations of these constructions have proved useful in various graph theory problems, but especially in many extremal problems which deal with cycles in graphs. I will explain motivations for these constructions, survey both old and new results, and state open questions.

This talk is part of the Combinatorics Seminar series.

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