Clar numbers of leapfrog fullerenes - Graver, Jack E. (Discrete mathematical chemistry, 2025)
A fullerene is a 3-regular plane graph with only hexagonal and pentagonal faces. The Fries number of a fullerene G, F(G), is the maximum number of benzene rings over all possible Kekulé structures for G. The Clar number of G, C(G), is the maximum number of independent benzene rings possible over all possible Kekulé structures for G. In this paper, we show that for leapfrog fullerenes, any set of faces attaining the Clar number is a subset of faces attaining the Fries number. This property is false for fullerenes in general (as shown in paper from E. J. Hartung in 2014). We then show that if L(G) is the leapfrog of a fullerene G, then the Clar number of L(G) is equal to the vertex independence number of G.
Keywords: chemical graph theory; Clar number; Clarovo število; Fries number; Friesovo število; fulereni; fullerenes; Kekulé structure; Kekuléjeva struktura; kemijska teorija grafov; leapfrog fullerenes; perfect matching; popolno prirejanje; preskakovalni fulereni