On The Ordering Of Trees By The Laplacian Coefficients

Let G = (V (G),E(G)) be a graph with n vertices, where V (G) and E(G) denote thevertex set and edge set of G. The Laplacian matrix of G is L(G) = D(G) ? A(G), whereD(G) is the diagonal matrix of vertex degrees and A(G) is the adjacency matrix of G. TheLaplacian polynomial of G is(?)The Laplacian matrix L(G) has non-negative eigenvalues (?)It is well known that c0(G) = 1, c1(G) = 2|E(G)|, cn(G) = 0, c_n-1(G) = nÏ„(G), whereÏ„(G)denotes the number of spanning trees of G.If G is a tree, coe?cient c_n-2(G) is equal to its Wiener index W(G), which is de?nedas the sum of distances between all pairs of vertices, that is,(?)In this thesis, we consider mainly the problems on ordering two types of trees by theirLaplacian coe?cients, Wiener index and Laplacian-like energy.First, we consider the trees with a given bipartition. Let T be a tree with n vertices.Hence its vertex set can be partitioned into two subsets V₁ and V₂, such that each edge joinsa vertex in V₁ with a vertex in V₂. Suppose that |V₁| = p, |V₂| = q, p≤q and p+q = n. Thenwe say that T has a (p,q)-bipartition. We characterize the trees with a given bipartition (p,q) which have the minimal and second minimal Laplacian coe?cients, Wiener index andLaplacian-like energy.Second, denote byψn the class of trees with n vertices, which have maximum degreeâ–³₁and second maximum degreeâ–³₂. We characterize the tree inψ_n which has the maximumLaplacian coefficients, Wiener index and Laplacian-like energy.