ON THE MULTIPLICATIVE ZAGREB COINDEX OF GRAPHS

For a (molecular) graph G with vertex set V (G) and edge set E (G) , the first and second Zagreb indices of G are defined as M-1(G) = Sigma upsilon is an element of V(G)d(G)(v())2 and M-2(G) = Sigma(uv)is an element of E(G) d(G)(u)d(G)(v) , respectively, where d G (v) is the degree of vertex v in G . The al-ternative expression of M-1(G) is Sigma E-uv is not an element of(G)(d(G)(u) + d(G)(v)) . Recently Ashrafi, Doslic' and Hamzeh introduced two related graphical invariants (M) over bar (1)(G) = Sigma E-uv is not an element of(G)(dG(u) vertical bar dG(v)) and (M) over bar (2)(G) = Sigma E-uv is not an element of(G)(d(G)(u)d(G)(v)) named as first Zagreb coindex and second Zagreb coindex, respectively. Here we define two new graphical invariants (pi) over bar (1)(G) = pi(uv is an element of E(G))(d(G)(u)d(G)(v)) and (pi) over bar (2)(G) = pi(uv is not an element of E(G))(dG(u)dG(v)) as the respective multiplicative versions of M i for i = 1 , 2 . In this paper, we have reported some properties, especiall y upper and lower bounds, for these two graph invariants of connected (molecular) gra phs. Moreover, some correspond-ing extremal graphs have been characterized with respect to these two indices.