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GraphTheory

 SpanningPolynomial

 Calling Sequence SpanningPolynomial(G, x)

Parameters

 G - undirected unweighted graph x - variable or value

Description

 • The SpanningPolynomial command returns a polynomial in x when x is a variable or the evaluation of the polynomial when x is a value. The value of this polynomial at a value $0\le p\le 1$ gives the probability that G is spanning (connected if G is connected) when each edge operates with probability p.

Examples

 > $\mathrm{with}\left(\mathrm{GraphTheory}\right):$
 > $\mathrm{with}\left(\mathrm{SpecialGraphs}\right):$
 > $G≔\mathrm{PetersenGraph}\left(\right):$
 > $f≔\mathrm{SpanningPolynomial}\left(G,x\right)$
 ${f}{≔}{704}{}{{x}}^{{15}}{-}{4920}{}{{x}}^{{14}}{+}{14430}{}{{x}}^{{13}}{-}{22755}{}{{x}}^{{12}}{+}{20370}{}{{x}}^{{11}}{-}{9828}{}{{x}}^{{10}}{+}{2000}{}{{x}}^{{9}}$ (1)
 > $\genfrac{}{}{0}{}{f}{\phantom{x=0.75}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{f}}{x=0.75}$
 ${0.8153727651}$ (2)
 > $\mathrm{SpanningPolynomial}\left(G,0.35\right)$
 ${0.02142980972}$ (3)