Research On The Laplacian Coefficients Of Graphs And Relative Spectral Theory

This paper which stands on the basis of previous results, do further research on sharp bounds of Laplacian coefficients and signless Laplacian index of some kinds of graphs. And the lexicographic ordering of unicyclic graphs with given diameter d(1≤d≤n－2) by spectral moments is considered. The main results as follows:· In the first two sections, we introduce the background and significance of the research, including the development of a representative at home and abroad regarding this aspect. Based on this research background and profound discus-sion on the status quo, it fully shows the main work’s necessity and innovation.· Ilic [10] characterized n-vertex trees with given matching number q which si-multaneously minimize all Laplacian coefficients. In Subsection3.3, we give another proof of this result. Using our method, we can go further beyond Ilic by giving the n-vertex tree with given matching number q which simul-tancously makes all Laplacian coefficients be the second smallest. We also determine the n-vertex trees with a perfect matching having the largest and second largest Laplacian coefficients, respectively. In Subsection3.5, extremal values on some indices, such as Wiener index, modified hyper-Wiener index, Laplacian-like energy, incidence energy of n-vertex trees with matching num-ber q are obtained.· Stevanovic, Ilic [D. Stevanovic, A. Ilic, On the Laplacian coefficients of uni-cyclic graphs, Linear Algebra Appl.430(2009)2290-2300] and He, Shan [C.X. He, H.Y. Shan, On the Laplacian coefficients of bicyclic graphs, Discrete Math.310(2010)3404-3412], identified graphs with the minimal Laplacian coefficients in the set of all n-vertex connected unicyclic graphs and bicyclic graphs, respectively. In Subsection3.4, using a more simpler method, we characterize the unicyclic (resp. bicyclic) graph with the minimal Laplacian coefficients. Furthermore, we use this unified approach to determine the tri-cyclic graph with the minimal Laplacian coefficients in the set of all n-vertex tricyclic graphs. A relation to the recently established Laplacian-like energy of a graph is discussed in Subsection3.5.· In Section4, we study the signless Laplacian spectral radius of unicyclic graphs with n vertices and diameter d. We determine graphs with the largest signless Laplacian spectral radius among all the unicyclic graphs with n vertices of diameter d. Moreover, if4≤d≤n－3with d≡0(mod2), then we identify unicyclic graphs with n vertices of diameter d having the second largest Q-index.· In Section5, the lexicographic ordering of unicyclic graphs with n vertices and diameter d(1≤d≤n－2) by spectral moments is considered, and the last4d－8unicyclic graphs, in an S-order, among all unicyclic graphs with n vertices and diameter d for4≤d≤n－8are identified. Moreover, we show that all unicyclic graphs with n vertices and diameter d have an S-order for d≤2and d≥n－2, respectively.