Resistance distance in straight linear 2-trees

We consider the graph Gn with vertex set V(Gn)={1,2,…,n} and {i,j}∈E(Gn) if and only if 0<|i−j|≤2. We call Gn the straight linear 2-tree on n vertices. Using Δ–Y transformations and identities for the Fibonacci and Lucas numbers we obtain explicit formulae for the resistance distance rGn(i,j) between any two vertices i and j of Gn. To our knowledge {Gn}n=3∞ is the first nontrivial family with diameter going to ∞ for which all resistance distances have been explicitly calculated. Our result also gives formulae for the number of spanning trees and 2-forests in a straight linear 2-tree. We show that the maximal resistance distance in Gn occurs between vertices 1 and n and the minimal resistance distance occurs between vertices n∕2 and n∕2+1 for n even (with a similar result for n odd). It follows that rn(1,n)→∞ as n→∞. Moreover, our explicit formula makes it possible to order the non-edges of Gn exactly according to resistance distance, and this ordering agrees with the intuitive notion of distance on a graph. Consequently, Gn is a geometric graph with entirely different properties than the random geometric graphs investigated in Luxburg, Radl and Hein (2010). These results for straight linear 2-trees along with an example of a bent linear 2-tree and empirical results for additional graph classes convincingly demonstrate that resistance distance should not be discounted as a viable method for link prediction in geometric graphs.