Chordal graphs and the characteristic polynomial
The greedoid ``characteristic polynomial'' is studied for greedoids (anitmatroids, in this case) which arise from chordal graphs. The greedoid two-variable ``Tutte polynomial'' of Gordon and McMahon is, for a greedoid on \(E\) having rank function \(r\), \[ f(t,z) = \sum_{S \subseteq E} t^{r(E) - r(S)} z^{|A|- r(A)}. \] The greedoid ``characteristic polynomial'' is then \(p(\lambda) = (-1)^{r(G)}f(-\lambda, -1)\) (not to be confused with the one-variable ``Tutte polynomial'' \(\lambda(t) = f(0, t-1)\) of Björner, Korte, and Lovász). This paper notes that, in the case of the chordal graphs \(G\), the greedoid characteristic polynomial is given as an alternating sum \((-1)^{|E|}\sum (-1)^k a_k \lambda^k\), where \(a_k\) is the number of cliques of \(G\) having \(k\) vertices. It is shown that, for connected \(G\), the value of the ``\(\beta\) invariant'' of the greedoid, defined as \(|p^\prime(1)|\), is the number of blocks of the graph.
scientific article; zbMATH DE number 1901901