Crossing numbers of Sierpiński‐like graphs

Abstract

The crossing number of Sierpiński graphs S(n, k) and their regularizations S + (n, k) and S ++ (n, k) is studied. Explicit drawings of these graphs are presented and proved to be optimal for S + (n, k) and S ++ (n, k) for every n ≥ 1 and k ≥ 1. These are the first nontrivial families of graphs of "fractal" type whose crossing number is known.

Maximum matchings in scale-free networks with identical degree distribution

“…For the second network, by employing the Pfaffian method proposed independently by Kasteleyn [32], Fisher and Temperley [33], we construct a Pfaffian orientation of the network. On the basis of Pfaffian orientation, we determine the number of perfect matchings as well as its entropy, which is proved equal to that corresponding to the extended Sierpiński graph [34]. Our findings suggest that the power-law degree distribution by itself cannot determine the properties of maximum matchings in scale-free networks.…”

“…Moreover, the number of perfect matchings in H g is very high, the entropy of which is equivalent to that of the extended Sierpiński graph [34], as will be shown below.…”

“…Both F g and H g have the same degree sequence for all g ≥ 1, but H g has perfect matchings. Moreover, we determine the number of perfect matchings in H g by applying the Pfaffian method, and show that H g has the same entropy for perfect matchings as that associated with an extended Sierpiński graph [34].…”

“…The extended Sierpiński graph is a particular case of Sierpiński-like graphs proposed by Klavžar and Moharin [34], which is in fact a variant of the Tower of Hanoi graph [53,54]. The extended Sierpiński graph can be defined by iteratively applying the subdivided-line graph operation [55].…”

On distances in Sierpiński graphs: Almost-extreme vertices and metric dimension

“…An appealing application of Sierpiński graphs is due to Romik [23] who designed, based on Sierpiński labelings, a finite automaton particularly useful for the Tower of Hanoi problem. In [19] the structure of Sierpiński graphs was the key to determine for the first time the exact genus of infinite families of fractal graphs. Recently, the hub number of Sierpiński-like graphs was determined in [15].…”