Related papers: The Laplacian eigenvalue 2 of bicyclic graphs

The Laplacian matrix of a graph $G$ is denoted by $L(G)=D(G)-A(G)$, where $D(G)=diag(d(v_{1}),\ldots , d(v_{n}))$ is a diagonal matrix and $A(G)$ is the adjacency matrix of $G$. Let $G_1$ and $G_2$ be two graphs. A one-edge connection of…

A signless Laplacian eigenvalue of a graph $G$ is called a main signless Laplacian eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this paper, all connected bicyclic graphs with exactly two main…

We review the properties of eigenvectors for the graph Laplacian matrix, aiming at predicting a specific eigenvalue/vector from the geometry of the graph. After considering classical graphs for which the spectrum is known, we focus on…

The Laplacian matrix of a simple graph is the difference of the diagonal matrix of vertex degree and the (0,1) adjacency matrix. In the past decades, the Laplacian spectrum has received much more and more attention, since it has been…

The signless Laplacian matrix of a graph $G$ is defined to be the sum of its adjacency matrix and degree diagonal matrix, and its eigenvalues are called $Q$-eigenvalues of $G$. A $Q$-eigenvalue of a graph $G$ is called a $Q$-main eigenvalue…

Denote the Laplacian of a graph $G$ by $L(G)$ and its second smallest Laplacian eigenvalue by $\lambda_2(G)$. If $G$ is a graph on $n\ge 2$ vertices, then it is shown that the second smallest eigenvalue of $L(G) + \frac{1}{n}…

Suppose that $G$ is a connected simple graph with the vertex set $V(G)=\{v_1, v_2,\cdots,v_n\}$. Then the adjacency matrix of $G$ is $A(G)=(a_{ij})_{n\times n}$, where $a_{ij}=1$ if $v_i$ is adjacent to $v_j$, and otherwise $a_{ij}=0$. The…

Consider a semigraph $G=(V,\,E)$; in this paper, we study the eigenvalues of the Laplacian matrix of $G$. We show that the Laplacian of $G$ is positive semi-definite, and $G$ is connected if and only if $\lambda_2 >0.$ Along the similar…

A signless Laplacian eigenvalue of a graph $G$ is called a main signless Laplacian eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this paper, we first give the necessary and sufficient conditions for a…

Let G be a graph on n vertices. The Laplacian matrix of G, denoted by L(G), is defined as L(G) = D(G) - A(G), where A(G) is the adjacency matrix of G and D(G) is the diagonal matrix of the vertex degrees of G. A graph G is said to be…

The parameter $\sigma(G)$ of a graph $G$ stands for the number of Laplacian eigenvalues greater than or equal to the average degree of $G$. In this work, we address the problem of characterizing those graphs $G$ having $\sigma(G)=1$. Our…

In this paper, we characterize all connected graphs with exactly three distinct normalized Laplacian eigenvalues of which one is equal to $1$, determine all connected bipartite graphs with at least one vertex of degree $1$ having exactly…

Let $G$ be a simple graph with adjacency matrix $A(G)$, signless Laplacian matrix $Q(G)$, degree diagonal matrix $D(G)$ and let $l(G)$ be the line graph of $G$. In 2017, Nikiforov defined the $A_\alpha$-matrix of $G$, $A_\alpha(G)$, as a…

We characterize all graphs for which there are eigenvectors of the graph Laplacian having all their components in {-1,+1} or {-1,0,+ 1}. Graphs having eigenvectors with components in {-1,+1} are called bivalent and are shown to be the…

For a simple connected graph $ G $ of order $ n $, the normalized Laplacian is a square matrix of order $ n $, defined as $\mathcal{L}(G)= D(G)^{-\frac{1}{2}}L(G)D(G)^{-\frac{1}{2}}$, where $ D(G)^{-\frac{1}{2}} $ is the diagonal matrix…

An eigenvalue of a graph $G$ is called a main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this paper, all connected tricyclic graphs with exactly two main eigenvalues are determined.

An eigenvalue of a graph $G$ is called a main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this paper, all connected tricyclic graphs with exactly two main eigenvalues are determined.

Let $G$ be a simple connected graph with vertex set $V(G)=\{v_{1}, v_{2}, \ldots, v_{n}\}$. The distance $d_G(v_i,v_j)$ between two vertices $v_i$ and $v_j$ of $G$ is the length of a shortest path between $v_i$ and $v_j$. The distance…

We present sharp inequalities relating the number of vertices, edges, and triangles of a graph to the smallest eigenvalue of its adjacency matrix and the largest eigenvalue of its Laplacian.

Graphs with few distinct eigenvalues have been investigated extensively. In this paper, we focus on another relevant topic: characterizing graphs with some eigenvalue of large multiplicity. Specifically, the normalized Laplacian matrix of a…