Some Properties of the Zagreb Indices

Let G = (V, E), V = {1, 2, ... , n}, E = {e(1), e(2), ... , e(m)}, be a simple graph with n vertices and m edges. Denote by d(1) >= d(2) >= ... >= d(n) > 0, and d(e(1)) >= d(e(2)) >= ... >= d(e(m)), sequences of vertex and edge degrees, respectively. If i-th and j-th vertices of G are adjacent, it is denoted as i similar to j. Graph invariants referred to as the first, second and the first reformulated Zagreb indices are defined as M-1 = Sigma(n)(i=1) d(i)(2), M-2 = Sigma(i similar to j) d(i)d(j) and EM1 = Sigma(m)(i=1) d(e(i))(2), respectively. Let lambda(1) >= lambda(2) >= ... >= lambda(n) be eigenvalues of G. With rho(G) = lambda(1) a spectral radius of G is denoted. Lower bounds for invariants M-1, M-2, EM1 and rho(G) are obtained.