LOWER BOUNDS FOR ENERGY OF MATRICES AND ENERGY OF REGULAR GRAPHS.

Let A = [aij ] be an n × n real symmetric matrix with eigenvalues λ₁,..., λ_n. The energy of A, denoted by E(A), is defined as |λ₁| + · · · + |λn|. We prove that if A is non-zero and |λ₁| ≥ · · · ≥ |λn|, then E(A) ≥ n|λ₁||λ_n| + P 1≤i,j≤n a 2 ij |λ₁| + |λ_n| (0.1). In particular, we show that Ψ(A)E(A) ≥ P 1≤i,j≤n a 2 ij, where Ψ(A) is the maximum value of the sequence Pn j=1 |a1j |, Pn j=1 |a2j |, . . . ,Pn j=1 |anj |. The energy of a simple graph G, denoted by E(G), is defined as the energy of its adjacency matrix. As an application of inequality (0.1) we show that if G is a t-regular graph (t ̸= 0) of order n with no eigenvalue in the interval (-1, 1), then E(G) ≥ 2tn t+1 and the equality holds if and only if every connected component of G is the complete graph K_t+1 or the crown graph K*_t+1. [ABSTRACT FROM AUTHOR]