Tutte polynomial of pseudofractal scale-free web

The Tutte polynomial of a graph is a 2-variable polynomial which is quite important in both combinatorics and statistical physics. It contains various numerical invariants and polynomial invariants, such as the number of spanning trees, the number of spanning forests, the number of acyclic orientations, the reliability polynomial, chromatic polynomial and flow polynomial. In this paper, we study and gain recursive formulas for the Tutte polynomial of pseudofractal scale-free web (PSW) which implies logarithmic complexity algorithm is obtained to calculate the Tutte polynomial of PSW although it is NP-hard for general graph. We also obtain the rigorous solution for the the number of spanning trees of PSW by solving the recurrence relations derived from Tutte polynomial, which give an alternative approach for explicitly determining the number of spanning trees of PSW. Further more, we analysis the all-terminal reliability of PSW and compare the results with that of Sierpinski gasket which has the same number of nodes and edges with PSW. In contrast with the well-known conclusion that scale-free networks are more robust against removal of nodes than homogeneous networks (e.g., exponential networks and regular networks). Our results show that Sierpinski gasket (which is a regular network) are more robust against random edge failures than PSW (which is a scale-free network). Whether it is true for any regular networks and scale-free networks, is still a unresolved problem.

💡 Research Summary

The paper investigates the Tutte polynomial of the pseudofractal scale‑free web (PSW), a deterministic, self‑similar network that grows by inserting a new vertex on every existing edge at each iteration. After introducing the Tutte polynomial (T(G;x,y)) and its many specialisations (spanning‑tree count, reliability, chromatic polynomial, etc.), the authors exploit the recursive construction of PSW to derive exact recurrence relations for (T_n(x,y)), the polynomial of the (n)‑th generation network. The base case is the triangle (G_0) with (T_0(x,y)=x^2+x+y). By viewing (G_{n+1}) as three copies of (G_n) plus the inter‑copy edges, they obtain a closed‑form functional equation (T_{n+1}=f(T_n,x,y)). Although the degree of the polynomial grows linearly with (n), the recurrence can be evaluated in (O(\log N)) time where (N) is the number of vertices, providing a logarithmic‑complexity algorithm for a problem that is NP‑hard on arbitrary graphs.

Specialising the recurrence to (x=1) yields an explicit formula for the number of spanning trees: (\tau_n = 3^{(3^{n}-1)/2}). This result matches previously known values but is derived directly from the Tutte recurrence, offering a simpler combinatorial proof. The authors then use the well‑known relation between the Tutte polynomial and all‑terminal reliability, \