Abstract. A signed graph Γ = (G, σ) consists of an unsigned graph G = (V, E) and a mapping σ : E → {+,−}. Let Γ be a connected signed graph and L(Γ),L(Γ) be its Laplacian matrix and normalized Laplacian matrix, respectively. Suppose μ1 ≥ · · · ≥ μn−1 ≥ μn ≥ 0 and λ1 ≥ · · · ≥ λn−1 ≥ λn ≥ 0 are the Laplacian eigenvalues and the normalized Laplacian eigenvalues of Γ, respectively. In this paper, ...

Let $D$ be a diameter and $d_G(v_i, v_j)$ be the distance between the vertices $v_i$ and $v_j$ of a connected graph $G$. The complementary distance signless Laplacian matrix of a graph $G$ is $CDL^+(G)=[c_{ij}]$ in which $c_{ij}=1+D-d_G(v_i, v_j)$ if $ineq j$ and $c_{ii}=sum_{j=1}^{n}(1+D-d_G(v_i, v_j))$. The complementary transmission $CT_G(v)$ of a vertex $v$ is defined as $CT_G(v)=sum_{u in ...

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

Suppose μ1, μ2, ... , μn are Laplacian eigenvalues of a graph G. The Laplacian energy of G is defined as LE(G) = ∑n i=1 |μi − 2m/n|. In this paper, some new bounds for the Laplacian eigenvalues and Laplacian energy of some special types of the subgraphs of Kn are presented. AMS subject classifications: 05C50

Graph embeddings are useful in bounding the smallest nontrivial eigenvalues of Laplacian matrices from below. For an n×n Laplacian, these embedding methods can be characterized as follows: The lower bound is based on a clique embedding into the underlying graph of the Laplacian. An embedding can be represented by a matrix Γ; the best possible bound based on this embedding is n/λmax(Γ Γ). Howeve...

Let G be a connected simple graph whose Laplacian eigenvalues are 0 = μn (G) μn−1 (G) · · · μ1 (G) . In this paper, we establish some upper and lower bounds for the algebraic connectivity and the largest Laplacian eigenvalue of G . Mathematics subject classification (2010): 05C50, 15A18.

Let G = (V,E) be a graph without loops and multiple edges. Let n and m be the number of vertices and edges of G, respectively. Such a graph will be referred to as an (n,m)-graph. For v ∈ V (G), let d(v) be the degree of v. In this paper, we are concerned only with undirected simple graphs (loops and multiple edges are not allowed). Let G be a graph with n vertices and the adjacency matrix A(G)....