Zagreb Indices Of Some Transformation Graphs

In 1972, Gutman and Trinajestic defined the first Zagreb index M1 and the secondZagreb index M2. For a given connected graph G, the first Zagreb index M1 is equal tothe sum of the squares of the degrees of the vertices, and the second Zagreb index M2 isequal to the sum of the products of the degrees of pairs of adjacent vertices.For a given connected graph G, five subdivision-related operators on G are definedin [D.M. Cvetkoci′c, M. Doob and H. Sachs, Spectra of Graphs-Theory and Applications,Academic Press, New York, 1980], and the resulting graphs are respectively denotedby L(G),S(G),R(G),Q(G) and T(G). Recently, using the latter four graphs, in [M.Eliasi, B. Taeri, Four new sums of graphs and their Wiener indices, Discr. Appl. Math.157(2009)794-803] the F-sums graphs are defined and the Wiener indices of the resultinggraphs are studied. In [ M.H. Khalifeh, H. Yousefi-Azari, A.R. Ashrafi, S.G. Wagner,Some new results on distance-based graph invariants, European J. Comb. 30(2009) 1149-1163], Khalifeh et al. presented exact formulae for the first and second Zagreb indices ofvarious graph operations including Cartesian product, composition, join, disjunction andsymmetric di?erence of graphs.In this paper, we supply exact formulae for the first and second Zagreb indices of thefive subdivision-related graphs,F–sums graphs and four products(Kronecker product,strong product, skew product and converse skew product).