General Position Polynomials
Vesna Iršič, Sandi Klavžar, Gregor Rus, James Tuite
Abstract
Abstract A subset of vertices of a graph G is a general position set if no triple of vertices from the set lie on a common shortest path in G . In this paper we introduce the general position polynomial as $$\sum _{i \ge 0} a_i x^i$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mo>∑</mml:mo> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msup> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msup> </mml:mrow> </mml:math> , where $$a_i$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> is the number of distinct general position sets of G with cardinality i . The polynomial is considered for several well-known classes of graphs and graph operations. It is shown that the polynomial is not unimodal in general, not even on trees. On the other hand, several classes of graphs, including Kneser graphs K ( n , 2), with unimodal general position polynomials are presented.