In Lecture 10, we introduced a fundamental object of spectral graph theory: the graph Laplacian, and established some of its basic properties. We then focused on the task of estimating the value of eigenvalues of Laplacians. In particular, we proved the Courant-Fisher theorem that is instrumental in obtaining upper-bounding estimates on eigenvalues. Today, we continue by showing a technique – s...

In this paper an efficient analytical method is presented for calculating the eigenvalues of special matrices related to finite element meshes (FEMs) with regular topologies. In the proposed method, a skeleton graph is used as the model of a FEM. This graph is then considered as the Cartesian product of its generators. The eigenvalues of the Laplacian matrix of the entire graph are then easily ...

Let µ 1 (G) ≥. .. ≥ µn (G) be the eigenvalues of the adjacency matrix of a graph G of order n, and G be the complement of G. Suppose F (G) is a fixed linear combination of µ i (G) , µ n−i+1 (G) , µ i G ¡ , and µ n−i+1 G ¡ , 1 ≤ i ≤ k. It is shown that the limit lim n→∞ 1 n max {F (G) : v (G) = n} always exists. Moreover, the statement remains true if the maximum is taken over some restricted fa...

New lower bounds for eigenvalues of a simple graph are derived. Upper and lower bounds for eigenvalues of bipartite graphs are presented in terms of traces and degree of vertices. Finally a non-trivial lower bound for the algebraic connectivity of a connected graph is given.

Using Lotker’s interlacing theorem on the Laplacian eigenvalues of a graph in [5] and Wang and Belardo’s interlacing theorem on the signless Laplacian eigenvalues of a graph in [6], we in this note obtain spectral conditions for some Hamiltonian properties of graphs. 2010Mathematics Subject Classification : 05C50, 05C45

For a simple connected graph G with n-vertices having Laplacian eigenvalues μ1, μ2, . . . , μn−1, μn = 0, and signless Laplacian eigenvalues q1, q2, . . . , qn, the Laplacian-energy-like invariant(LEL) and the incidence energy (IE) of a graph G are respectively defined as LEL(G) = ∑n−1 i=1 √ μi and IE(G) = ∑n i=1 √ qi. In this paper, we obtain some sharp lower and upper bounds for the Laplacian...

A graph can be associated with a matrix in several ways. For instance, by associating the vertices of the graph to the rows/columns and then using 1 to indicate an edge and 0 otherwise we get the adjacency matrix A. The combinatorial Laplacian matrix is defined by L = D − A where D is a diagonal matrix with diagonal entries the degrees and A is again the adjacency matrix. Both of these matrices...