Lower bounds for the algebraic connectivity of graphs with specified subgraphs

Abstract

The second smallest eigenvalue of the Laplacian matrix of a graph G is called the algebraic connectivity and denoted by a ( G ) . We prove that a ( G )> π 2 /3( p (12 g ( n 1 , n 2 , …, n p ) 2 − π 2 )/4 g ( n 1 , n 2 , …, n p ) 4 + 4( q − p )(3 g ( n p + 1 , n p + 2 , …, n q ) 2 − π 2 )/ g ( n p + 1 , n p + 2 , …, n q ) 4 ), holds for every non-trivial graph G which contains edge-disjoint spanning subgraphs G 1 , G 2 , …, G q such that, for 1 ≤ i ≤ p , a ( G i )≥ a ( P n i ) , with n i ≥ 2 , and, for p + 1 ≤ i ≤ q , a ( G i )≥ a ( C n i ) , where P n i and C n i denote the path and the cycle of the corresponding order, respectively, and g denotes the geometric mean of given arguments. Among certain consequences, we emphasize the following lower bound a ( G )> π 2 12(4 q − 3 p ) n 2 − (16 q − 15 p ) π 2 /12 n 4 , referring to G which has n ( n ≥ 2 ) vertices and contains p Hamiltonian paths and q − p Hamiltonian cycles, such that all of them are edge-disjoint. We also discuss the quality of the obtained lower bounds.