Related papers: Laplacian integral graphs with a given degree sequ…

In this paper, structural properties of chordal graphs are analysed, in order to establish a relationship between these structures and integer Laplacian eigenvalues. We present the characterization of chordal graphs with equal vertex and…

In this note, we give an infinite family of optimal graphs called $G^+(d,c)$. They are optimal in the sense that they have the maximum possible number of vertices for given a diameter $d$ and the so-called `outer multiset dimension' $c$. We…

We study entanglement properties of mixed density matrices obtained from combinatorial Laplacians. This is done by introducing the notion of the density matrix of a graph. We characterize the graphs with pure density matrices and show that…

Let $M=[m_{ij}]$ be an $n\times m$ real matrix, $\rho$ be a nonzero real number, and $A$ be a symmetric real matrix. We denote by $D(M)$ the $n\times n$ diagonal matrix $diag(\sum_{j=1}^{m}m_{1j},\ldots,\sum_{j=1}^{m}m_{nj})$ and denote by…

Let $\Gamma = (G, \sigma)$ be a signed graph, where $G = (V(G),E(G))$ is an (unsigned) graph, called the underlying graph. The net Laplacian matrix of $\Gamma$ is defined as $L^{\pm}(\Gamma) = D^{\pm}(\Gamma) - A(\Gamma)$, where…

For a simple signed graph $G$ with the adjacency matrix $A$ and net degree matrix $D^{\pm}$, the net Laplacian matrix is $L^{\pm}=D^{\pm}-A$. We introduce a new oriented incidence matrix $N^{\pm}$ which can keep track of the sign as well as…

For a connected graph $G$, we denote by $L(G)$, $m_{G}(\lambda)$, $c(G)$ and $p(G)$ the line graph of $G$, the eigenvalue multiplicity of $\lambda$ in $G$, the cyclomatic number and the number of pendant vertices in $G$, respectively. In…

For a group $G$, the generating graph $\Gamma(G)$ is defined as the graph with the vertex set $G$, and any two distinct vertices of $\Gamma(G)$ are adjacent if they generate $G$. In this paper, we study the generating graph of $D_n,$ where…

Let $Z(G)$ be the centre of a finite non-abelian group $G.$ The non-commuting graph of $G$ is a simple undirected graph with vertex set $G\setminus Z(G),$ and two vertices $u$ and $v$ are adjacent if and only if $uv\ne vu.$ In this paper,…

Let $m_GI$ denote the number of Laplacian eigenvalues of a graph $G$ in an interval $I$ and let $\alpha(G)$ denote the independence number of $G$. In this paper, we determine the classes of graphs that satisfy the condition…

Recently normalized Laplacian matrices of graphs are studied as density matrices in quantum mechanics. Separability and entanglement of density matrices are important properties as they determine the nonclassical behavior in quantum…

Let $G$ be a simple graph with the Laplacian matrix $L(G)$ and let $e(G)$ be the number of edges of $G$. A conjecture by Brouwer and a conjecture by Grone and Merris state that the sum of the $k$ largest Laplacian eigenvalues of $G$ is at…

Graph embedding seeks to build a low-dimensional representation of a graph G. This low-dimensional representation is then used for various downstream tasks. One popular approach is Laplacian Eigenmaps, which constructs a graph embedding…

Let $G$ be a finite simple graph. The line graph $L(G)$ represents the adjacencies between edges of $G$. We define first the line simplicial complex $\Delta_L(G)$ of $G$ containing Gallai and anti-Gallai simplicial complexes…

For every real $0\leq \alpha \leq 1$, Nikiforov defined the $A_{\alpha}$-matrix of a graph $G$ as $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the degree diagonal matrix of a graph $G$,…

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of vertex degrees of $G$. For any real $\alpha \in [0,1]$, Nikiforov defined the $A_\alpha$-matrix of a graph $G$ as $A_\alpha(G)=\alpha…

The Laplacian spread of a graph is the difference between the largest eigenvalue and the second-smallest eigenvalue of the Laplacian matrix of the graph. We find that the class of strongly regular graphs attains the maximum of largest…

Let $G$ be a group and $L(G)$ be the set of all subgroups of $G$. We introduce a bipartite graph $\mathcal{B}(G)$ on $G$ whose vertex set is the union of two sets $G \times G$ and $L(G)$, and two vertices $(a, b) \in G \times G$ and $H \in…

For a graph \(G\) with no isolated vertices, its Laplacian ratio is defined as \[ \pi(G)=\frac{\operatorname{per}(L(G))}{\prod_{v\in V(G)} d(v)}, \] where \(L(G)\) is the Laplacian matrix of \(G\), \(d(v)\) is the degree of \(v\), and…

The signless Laplacian matrix of a graph $G$ is given by $Q(G)=D(G)+A(G)$, where $D(G)$ is a diagonal matrix of vertex degrees and $A(G)$ is the adjacency matrix. The largest eigenvalue of $Q(G)$ is called the signless Laplacian spectral…