LOWER BOUNDS FOR ENERGY OF MATRICES AND ENERGY OF REGULAR GRAPHS

Let A - [a(ij)] be an n x n real symmetric matrix with eigenvalues lambda(1), ..., lambda(n). The energy of A, denoted by E(A), is defined as vertical bar lambda(1)vertical bar + ... + vertical bar lambda(n)vertical bar . We prove that if A is non-zero and vertical bar lambda(1)vertical bar >= ... >= vertical bar lambda(n)vertical bar , then (0.1) E(A) >= n vertical bar lambda(1)vertical bar vertical bar lambda(n)vertical bar + Sigma(1 <= i,j <= n) a(ij)(2)/vertical bar lambda(1)vertical bar + vertical bar lambda(n)vertical bar. In particular, we show that Psi(A) E(A) >= Sigma(1 <= i,j <= n) a(ij)(2), where Psi(A) is the maximum value of the sequence Sigma(n)(j=1) vertical bar a(1j)vertical bar, Sigma(n)(j=1) vertical bar a(2j)vertical bar, ..., Sigma(n)(j=1) vertical bar a(nj)vertical bar. 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 not equal 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 Kt+1 or the crown graph K-t+1(star).