1. A new lower bound for the energy of graphs.
- Author
-
Oboudi, Mohammad Reza
- Subjects
- *
COMPLETE graphs , *EIGENVALUES - Abstract
Let G be a simple graph with n vertices v 1 , ... , v n and m edges. Let A (G) = [ a i j ] n × n be the adjacency matrix of G , that is a i j = 1 if v i and v j are adjacent, and a i j = 0 , otherwise. Let λ 1 , ... , λ n be the eigenvalues of A (G). The energy of G , denoted by E (G) , is defined as | λ 1 | + ⋯ + | λ n |. It is well known that 2 m ≤ E (G) ≤ 2 m n. In this paper we improve the latter lower bound and prove that if all eigenvalues of G are non-zero, then E (G) ≥ 2 | λ 1 λ n | | λ 1 | + | λ n | 2 m n , where | λ 1 | ≥ ⋯ ≥ | λ n |. We show that the equality holds if and only if G = n 2 K 2 or G = p K β + 1 ⋃ q T β + 1 , where p and q are some non-negative integers and β ≥ 2 and T β + 1 is the graph K β + 1 , β + 1 ∖ M , where M is a perfect matching of K β + 1 , β + 1. Finally by studying the | λ 1 λ n | | λ 1 | + | λ n | of graphs, we find some lower bounds for the energy of graphs. In particular, we obtain that if G has no eigenvalue in the interval (− 1 , 1) , then E (G) ≥ 2 m 2 n − 2 n , and the equality holds if and only if G is the complete graph K n. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF