Macaulay2 » Documentation
Packages » Oscillators :: vertexSpanningPolynomial
next | previous | forward | backward | up | index | toc

vertexSpanningPolynomial -- computes the vertex spanning polynomial

Description

Let $S$ be the set of all spanning trees of the graph $G$. For each spanning tree $T$, let $d_i$ be the degree of $v_i$ in $T$. The vertex spanning polynomial of a graph $G$ is defined as $\sum_{T\in S} \prod_{v_i \in T} x_i^{d_i-1}$. The factorization of this polynomial is related to the number of components of the oscillator ideal of the graph computed via oscQuadrics.

i1 : G = graph({0,1,2,3}, {{0,1},{1,2},{2,3},{0,3}});
i2 : vertexSpanningPolynomial G

o2 = x x  + x x  + x x  + x x
      0 1    1 2    0 3    2 3

o2 : QQ[x ..y ]
         0   3

See also

Ways to use vertexSpanningPolynomial:

  • vertexSpanningPolynomial(Graph)
  • vertexSpanningPolynomial(Graph,Ring)

For the programmer

The object vertexSpanningPolynomial is a method function.


The source of this document is in /build/reproducible-path/macaulay2-1.25.05+ds/M2/Macaulay2/packages/Oscillators/Documentation.m2:472:0.