The Four Topological Indices Of L_n,p*and Trees With The Second-Largest（Smallest）,Third-Largest（Smallest）Laplacian Coefficients

The spectra of a graph, the Wiener index, the Hosoya index and Merrifield-Simmons index are typical examples of graph invariants in theoretical chemistry for quantifying relevant details of molecular structure. Recently, the extremal problem on these indices has been investigated extensively, i.e. determing the graphs with maximal or minimal topical indices. Let A(G) be the adjacency matrix of G and (?)φ(G; x) be its characteristic polynomial. The adjacency spectrum of G consists of the adjacency eigenvalues, which are the roots of the equation φ(G; x)=0. The spectral radius ρ(G) is the largest adjacency eigenvalue. The Wiener index of G is denned as W(G)=∑{u,v} dG(u,v), where dG(u,v) is the distance between u and v in G and the sum goes over all pairs of vertices. The Hosoya index is defined as Z(G)=∑k≥m(G; k), where m(G; k) is the number of k-independent edges sets of G, m(G;0)=1; The Merrifield-Simmons index is defined as i(G)=∑k≥0i(G; k), where i(G; k) is the number of k-independent vertex sets of G, i(G;0)=1. In Chapter2of this paper, we discuss the graph of Ln,p*with minimal, maximal, second-smallest, second-largest spectra radius and the extremal graph with minimal, maximal Wiener index, Hosoya index and Merrifield-Simmons index by four graph transformations.Let G be a simple undirected graph of order n with the Laplacian matrix L(G), P(G,λ)=∑kn=0ckλn-k.It is proved that among all trees of n vertices, the kth coefficient ck is largest when the tree is a path, and is smallest for stars. In Chapter3of this paper, we characterize the trees which have the second-largest (-smallest), third-largest (-smallest) Laplacian coefficient ck.