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.

$\mathrm{with}\left(\mathrm{GraphTheory}\right)\:$

$\mathrm{with}\left(\mathrm{SpecialGraphs}\right)\:$

$G≔\mathrm{PetersenGraph}\left(\right)\:$

$f≔\mathrm{SpanningPolynomial}\left(G\,x\right)$

$f≔704x^{15}-4920x^{14}+14430x^{13}-22755x^{12}+20370x^{11}-9828x^{10}+2000x^9$

$\mathrm{eval}\left(f\,x=0.75\right)$

$0.8153727651$

$\mathrm{SpanningPolynomial}\left(G\,0.35\right)$

$0.02142980972$