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# On the zeros of some genus polynomials

Published:1997-06-01
Printed: Jun 1997
• Saul Stahl
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## Abstract

In the genus polynomial of the graph $G$, the coefficient of $x^k$ is the number of distinct embeddings of the graph $G$ on the oriented surface of genus $k$. It is shown that for several infinite families of graphs all the zeros of the genus polynomial are real and negative. This implies that their coefficients, which constitute the genus distribution of the graph, are log concave and therefore also unimodal. The geometric distribution of the zeros of some of these polynomials is also investigated and some new genus polynomials are presented.
 MSC Classifications: 05C10 - Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25] 05A15 - Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx] 30C15 - Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) {For algebraic theory, see 12D10; for real methods, see 26C10} 26C10 - Polynomials: location of zeros [See also 12D10, 30C15, 65H05]