This is the third part of our work with a common title. The first [11] and the second part [12] will be also referred in the sequel as Part I and Part II, respectively. This third part was not planned at the beginning, but a lot of recently published papers on the signless Laplacian eigenvalues of graphs and some observations of ours justify its preparation. By a spectral graph theory we unders...

We study a general class of recurrence relations that appear in the application matrix diagonalization procedure. find closed formula and determine analytical properties solutions. finally apply these findings several problems involving eigenvalues graphs.

Let $G = (V, E)$ be a simple graph. Denote by $D(G)$ the diagonal matrix $diag(d_1,cdots,d_n)$, where $d_i$ is the degree of vertex $i$  and  $A(G)$ the adjacency matrix of $G$. The  signless Laplacianmatrix of $G$ is $Q(G) = D(G) + A(G)$ and the $k-$th signless Laplacian spectral moment of  graph $G$ is defined as $T_k(G)=sum_{i=1}^{n}q_i^{k}$, $kgeqslant 0$, where $q_1$,$q_2$, $cdots$, $q_n$ ...

In this thesis we aim to develop a framework for graph characterization by combining the methods from spectral graph theory and manifold learning theory. The algorithms are applied to graph clustering, graph matching and object recognition. Spectral graph theory has been widely applied in areas such as image recognition, image segmentation, motion tracking, image matching and etc. The heat kern...

This paper is dedicated to present a proof of the Spectral Theorem, and to discuss how the Spectral Theorem is applied in combinatorics and graph theory. In this paper, we also give insights into the ways in which this theorem unveils some mysteries in graph theory, such as expander graphs and graph coloring.