The Normalized Laplacian Spectrum, Degree-kirchhoff Index And The Number Of Spanning Trees Of Linear Pentagonal Chains

The spectral theory is an important field of graph theory and combinatorial matrix theory. It mainly establishes the topological structure of the graph by relevant matrices (adjacency matrix, Laplacian matrix, signless Laplacian matrix and so on),and studies the relationship between the invariants of the graph and the spectra of these relevant matrices.The normalized Laplacian of a graph, which is consistent with the matrix in spectral geometry and random walks, is a hot topic in recent years. And in many cases, the parameters based on this matrix can more faithfully reflect the structure and properties than other matrices of a graph. Thus, the normalized Laplacian has attracted more and more researchers' attention.Let Pn be a linear pentagonal chain with 2n pentagons. We get the chain by con-necting the corresponding vertices vi,vi' of two paths respectively whose vertex sets are {1,2,...,2n+1} and {1',2',...,(2n+1)'} to get edges vivi'=1,2,3,...,2n+1,and then subdividing edges vjvj ,j= 2,4,6,..., 2n. In this article, we get the rela-tionship between the roots and coefficients of the characteristic polynomials of the normalized Laplacian matrix and then study the degree-Kirchhoff index and spanning trees of Pn . The concrete content is in the following:· In Chapter 1, we give some necessary definition and introduce the background of the research and existing research results.· In Chapter 2, the normalized Laplacian polynomial decomposition of Pn and two important lemmas are given.· In Chapter 3, according to the decomposition theorem of normalized Laplacian polynomial of Pn, we obtain that the normalized Laplacian spectrum of Pn consists of the eigenvalues of two special matrices: LA of order 3n + 1 and LS of order 2n.+1. Furthermore, together with the relationship between the roots and coefficients of the characteristic polynomials of the above two matrices,explicit closed-form formulas of the degree-Kirchhoff index and the number of spanning trees of Pn are derived. Finally, it is interesting to find that the degree-Kirchhoff index of Pn is approximately to one half of its Gutman index.· In Chapter 4, we give some prospects for further research in the future.