International Mathematical Forum, Vol. 7, 2012, no. 14, 669-673 More on Zagreb Coindices of Composite Graphs Maolin Wang and Hongbo Hua Faculty of Mathematics and Physics Huaiyin Institute of Technology Huai an, Jiangsu 223003, P.R. China mlw.math@gmail.com, hongbo hua@163.com Abstract For a nontrivial graph G, its first and second Zagreb coindices are defined [1], respectively, as (d G (u)+d G (v)) and M 2 (G) = d G (u)d G (v), where d G (x) is the degree of vertex x in G. In this paper, we obtained some new properties of Zagreb coindices. We mainly give explicit formulae for the first Zagreb coindex of line graphs and total graphs. Mathematics Subject Classification: 05C90; 05C35 05C12 Keywords: Degree; Zagreb indices; Zagreb coindices; line graphs; total graphs 1 Introduction Let G be a simple connected graph with vertex set V (G) and edge set V (G). For a graph G, we let d G (v) be the degree of a vertex v in G and let d G (u, v) denote the distance between vertices u and v in G. A graph invariant is a function defined on a graph which is independent of the labeling of its vertices. Till now, hundreds of different graphs invariants have been employed in QSAR/QSPR studies, some of which have been proved to be successful (see [9]). Among those successful invariants, there are two invariants called the first Zagreb index and the second Zagreb index (see [3, 5, 7, 8, 10 12]), defined as respectively. u V (G) (d G (u)) 2 and M 2 (G) = d G (u)d G (v),
670 Maolin Wang and Hongbo Hua In fact, one can rewrite the first Zagreb index as (d G (u)+d G (v)). (1) Noticing that contribution of nonadjacent vertex pairs should be taken into account when computing the weighted Wiener polynomials of certain composite graphs (see [4]), Ashrafi et al. [1, 2] defined the first Zagreb coindex and second Zagreb coindex as (d G (u)+d G (v)) and M 2 (G) = d G (u)d G (v), respectively. Ashrafi et al. [1] explored basic mathematical properties of Zagreb coindices and in particular presented explicit formulae for these new graph invariants under several graph operations, such as, union, join, Cartesian product, disjunction product, and so on. Ashrafi et al. [2] determined the extremal values of Zagreb coindices over some special classes of graphs. Hua and Zhang [6] revealed some relations between Zagreb coindices and some other distance-based topological indices. In this paper, we obtained some new properties of Zagreb coindices. We mainly give explicit formulae for the first Zagreb coindex of line graphs and total graphs. 2 M 1 index of line graphs and total graphs It is not difficult to see that the contribution of each vertex u in G to M 1 (G) is exactly (n d G (u) 1)d G (u). Thus, we have (n d G (u) 1)d G (u). (2) u V (G) Given a graph G, its line graph L(G) is a graph such that each vertex of L(G) represents an edge of G; and two vertices of L(G) are adjacent if and only if their corresponding edges share a common endpoint ( are adjacent ) in G. The total graph T (G) of a graph G is a graph such that the vertex set of T (G) corresponds to the vertices and edges of G and two vertices are adjacent in T (G) if and only if their corresponding elements are either adjacent or incident in G. For a positive integer, we let δ k (G) = (d G (v)) k. One can see that v V (G) δ 1 (G) is just the number of edges in G, and δ 2 (G) is just the first Zagreb index M 1 (G).
More on Zagreb coindices of composite graphs 671 Theorem 2.1 Let G be a nontrivial graph of order n and size m. Then M 1 (L(G)) = (m +3)M 1 (G) 2M 2 (G) 2m(m +1) δ 3 (G). Proof. Suppose that e i = u i v i (i =1,...,m) is a vertex in L(G). It can be easily seen that d L(G) (e i )=d G (u i )+d G (v i ) 2. Note that for a nonempty graph G, it holds [(d G (u)) 2 +(d G (v)) 2 ]= (d G (w)) 3 = δ 3 (G). By means of the equation (2), we have M 1 (L(G)) = (m d L(G) (e i ) 1)d L(G) (e i ) e i V (L(G)) w V (G) = [m (d G (u i )+d G (v i ) 2) 1](d G (u i )+d G (v i ) 2) u i v i E(G) =(m +3)M 1 (G) 2m(m +1) (d G (u i )+d G (v i )) 2 u i v i E(G) =(m +3)M 1 (G) 2M 2 (G) 2m(m +1) (d G (u i )) 3 u i V (G) =(m +3)M 1 (G) 2M 2 (G) 2m(m +1) δ 3 (G), as desired. Theorem 2.2 Let G be a nontrivial graph of order n and size m. Then M 1 (T (G)) = 4m(n + m 1) + (n + m 5)M 1 (G) 2M 2 (G) δ 3 (G). Proof. Note that T (G) has n + m vertices. By means of the equation (2), we have M 1 (T (G)) = = w V (T (G)) w V (T (G)) V (G) (n + m d T (G) (w) 1)d T (G) (w) w V (T (G)) E(G) (n + m d T (G) (w) 1)d T (G) (w)+ (n + m d T (G) (w) 1)d T (G) (w). Note that for w V (T (G)) V (G), d T (G) (w) =2d G (w), and for w = uv V (T (G)) E(G), d T (G) (w) =(d G (u)+d G (v) 2) + 2 = d G (u)+d G (v).
672 Maolin Wang and Hongbo Hua Hence, M 1 (T (G)) = (n + m 2d G (w) 1)2d G (w)+ w V (G) (n + m (d G (u)+d G (v)) 1)(d G (u)+d G (v)) w=uv V (T (G)) E(G) =2(n + m 1) 2m 4M 1 (G)+(n + m 1)M 1 (G) (d G (u)+d G (v)) 2 =4m(n + m 1) + (n + m 5)M 1 (G) 2M 2 (G) [(d G (u)) 2 + d G (v)) 2 ] =4m(n + m 1) + (n + m 5)M 1 (G) 2M 2 (G) δ 3 (G). This completes the proof. References [1] A.R. Ashrafi, T. Došlić, A. Hamzeha, The Zagreb coindices of graph operations, Discrete Appl. Math., 158 (2010) 1571-1578. [2] A.R. Ashrafi, T. Došlić, A. Hamzeha, Extremal graphs with respect to the Zagreb coindices, MATCH Commun. Math. Comput. Chem., 65 (2011) 85-92. [3] K.C. Das, I. Gutman, B. Zhou, New upper bounds on Zagreb indices, J. Math. Chem., 46 (2009) 514-521. [4] T. Došlić, Vertex-weighted Wiener polynomials for composite graphs, Ars Math. Contemp., 1 (2008) 66-80. [5] J. Hao, Theorems about Zagreb indices and modified Zagreb indices, MATCH Commun. Math. Comput. Chem., 65 (2011) 659-670. [6] H. Hua and S. Zhang, Relations between Zagreb coindices and some distance-based topological indices, MATCH Commun. Math. Comput. Chem., in press. [7] M.H. Khalifeh, H. Yousefi-Azari, A.R. Ashrafi, The first and second Zagreb indices of graph operations, Discrete Appl. Math., 157 (2009) 804-811.
More on Zagreb coindices of composite graphs 673 [8] B. Liu, Z. You, A survey on comparing Zagreb indices, MATCH Commun. Math. Comput. Chem., 65 (2011) 581-593. [9] R. Todeschini, V. Consoni, Handbook of Molecular Descriptors, Wiley- VCH, New York, 2002. [10] K. Xu, The Zagreb indices of graphs with a given clique number, Appl. Math. Lett., 24 (2011) 1026-1030. [11] B. Zhou, Zagreb indices, MATCH Commun. Math. Comput. Chem., 52 (2004) 113-118. [12] B. Zhou, Upper bounds for the Zagreb indices and the spectral radius of series-parallel graphs, Int. J. Qutum. Chem., 107 (2007) 875-878. Received: September, 2011