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arXiv · 1701.08564

On sequences of polynomials arising from graph invariants

T. Kotek·J. A. Makowsky·E. V. Ravve
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Abstract

Graph polynomials are deemed useful if they give rise to algebraic characterizations of various graph properties, and their evaluations encode many other graph invariants. Algebraic: The complete graphs $K_n$ and the complete bipartite graphs $K_{n,n}$ can be characterized as those graphs whose matching polynomials satisfy a certain recurrence relations and are related to the Hermite and Laguerre polynomials. An encoded graph invariant: The absolute value of the chromatic polynomial $χ(G,X)$ of a graph $G$ evaluated at $-1$ counts the number of acyclic orientations of $G$. In this paper we prove a general theorem on graph families which are characterized by families of polynomials satisfying linear recurrence relations. This gives infinitely many instances similar to the characterization of $K_{n,n}$. We also show where to use, instead of the Hermite and Laguerre polynomials, linear recurrence relations where the coefficients do not depend on $n$. Finally, we discuss the distinctive power of graph polynomials in specific form.

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T. Kotek, J. A. Makowsky, E. V. Ravve. 2017-03-02. On sequences of polynomials arising from graph invariants. https://arxiv.org/abs/1701.08564

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