On The Ordering Of Two Types Of Trees By Their Energies And Hosoya Indices

Suppose that G=(V,E) is a simple graph with vertex set V={v1,v2,…,vn} and edge setE={e1,e2,…,em}. The adjacency matrix of G is a square matrix of order n, denoted byA(G)=(aij)n , where aij equals 1 if the vertices vi and vj are adjacent and 0 otherwise. ×nDenote the characteristic polynomial of A(G) by φ(G )= det(xI ? A(G)), where I is a unitmatrix of order n. The zeros of φ(G) are called the eigenvalues of graph G. Particularly, if Tis a tree with n vertices, then [n/2] φ(T) = ∑ (?1)km(T, k)xn?2k , k =0where m(T,k) is the number of k-matchings of T(the number of matchings with k edges of T),m(T,0)=1 by convention. The energy of a graph G, denoted by E(G), is defined as the sum ofabsolute values of eigenvalues of G,that is, E(G)= λ1 + λ2 +...+ λn ,where λi, for i=1,2,…,n, are the eigenvalues of G. The number of matchings of graph G iscalled the Hosoya index, denoted by Z(G), i.e.: Z(G)=m(G,0)+m(G,1)+…+m(G,[n/2])。For a tree T with n vertices, the energy of T can be expressed in terms of the Coulsonintegral formula as follows: 2 +∞ [n/2] E(T)= x?2 ln[1+ m(T, k)x2 ]dx . k π ∫ ∑ 0 k=1We consider mainly the problems on ordering two types of trees by their energies andHosoya indices. First, the trees with a given bipartition are considered. It is well known that for anarbitrary bipartite graph G the set of vertices of G can be partitioned into two parts V1 and V2such that each edge of G joins one vertex in V1 and another in V2. Suppose that |V1|=p and|V2|=q, and q ≥ p , n=p+q. We say the bipartite graph G has a bipartition (p,q). Wecharacterize the trees with a given bipartition (p,q) which have the minimal and secondminimal energies and Hosoya indices.Second, suppose that n and p are two positive integers, n ≥ p +1≥ 3. We characterize the iitree with n vertices and with at most p pendant vertices, which has the minimal energy andHosoya index.