Anti-Forcing Spectra of Convex Hexagonal Systems

For any perfect matching M of a graph A G , the anti-forcing number of M a f ( G , M ) is the cardinality of a minimum edge subset S ⊆ E ( G ) \ M such that the graph G − S has only one perfect matching. The anti-forcing numbers of all perfect matchings of G form its anti-forcing spectrum, denoted by Spec af ( G ) . For a convex hexagonal system O ( n 1 , n 2 , n 3 ) with n 1 ≤ n 2 ≤ n 3 , denoted by H , it has the minimum anti-forcing number n 1 . In this paper, we derive a formula for its maximum anti-forcing number A f ( H ) , i.e., the Fries number. Next, we prove that [ n 1 , c ] ∪ { c + 2 , c + 4 , … , A f ( H ) − 2 , A f ( H ) } ⊆ Spec a f ( H ) for the specific integer c with the same parity as A f ( H ) . In particular, we obtain that if n 1 + n 2 − n 3 ≤ 1 , then c = A f ( H ) , which implies that Spec af ( H ) = [ n 1 , A f ( H ) ] is an integer interval. Finally, we also give some non-continuous situations: Spec af ( O ( 2 , n , n ) ) = [ 2 , 4 n − 2 ] \ { 4 n − 3 } for n ≥ 2 ; the anti-forcing spectrum of H has a gap A f ( H ) − 1 for n 1 = n 2 ≥ 2 and n 3 even, or n 2 = n 3 and n 1 ≥ 2 even.