Related papers: The signless Laplacian state transfer in Q-graph

Suppose that the vertex set of a connected graph $G$ is $V(G)=\{v_1,\cdots,v_n\}$. Then we denote by $Tr_{G}(v_i)$ the sum of distances between $v_i$ and all other vertices of $G$. Let $Tr(G)$ be the $n\times n$ diagonal matrix with its…

A continuous-time quantum walk on a graph is a matrix-valued function $\exp(-\mathtt{i} At)$ over the reals, where $A$ is the adjacency matrix of the graph. Such a quantum walk has universal perfect state transfer if for all vertices $u,v$,…

Let $G$ be a connected graph and $Q(G)$ be the signless Laplacian of $G$. The principal ratio $\gamma(G)$ of $Q(G)$ is the ratio of the maximum and minimum entries of the Perron vector of $Q(G)$. In this paper, we consider the maximum…

Perfect state transfer between two marked vertices of a graph by means of discrete-time quantum walk is analyzed. We consider the quantum walk search algorithm with two marked vertices, sender and receiver. It is shown by explicit…

The Helmholtzian matrix of a graph $G=(V(G),E(G))$ is a graph-theoretic analogue of the vector Laplacian (or Helmholtz operator) [S. Li, L. Lu, J.F. Wang, A graph discretization of vector Laplacian, 379 (2026) 446--460]. Motivated by the…

For any given graph G = (V,E) we define in a certain way a new graph G(x,y,z) with the vertex set V\cup E depending on parameters x,y,z from {0,1, +, -} and call graph G(x,y,z) the (x,y,z)-transformation of G. It turns out that if G is an…

A graph is determined by its signless Laplacian spectrum if there is no other non-isomorphic graph sharing the same signless Laplacian spectrum. Let $C_l$, $P_l$, $K_l$ and $K_{s,l-s}$ be the cycle, the path, the complete graph and the…

This paper investigates perfect state transfer in Grover walks, a model of discrete-time quantum walks. We establish a necessary and sufficient condition for the occurrence of perfect state transfer on graphs belonging to an association…

The \textit{transition matrix} of a graph $\Gamma$ with adjacency matrix $A$ is defined by $H(\tau ) := \exp(-\mathbf{i}\tau A)$, where $\tau \in \mathbb{R}$ and $\mathbf{i} = \sqrt{-1}$. The graph $\Gamma$ exhibits \textit{perfect state…

In a continuous-time quantum walk on a network of qubits, pretty good state transfer is the phenomenon of state transfer between two vertices with fidelity arbitrarily close to 1. We construct families of graphs to demonstrate that there is…

A spanning subgraph $F$ of a graph $G$ is defined as an even factor of $G$, if the degree $d_F(v)=2k, k\in\mathbb{N}^+$ for every vertex $v\in V(G)$. This note establishes a sufficient condition to ensure that a connected graph $G$ of even…

Let $G$ be a connected graph on $n$ vertices with girth $g$. Let $m_GI$ denote the number of Laplacian eigenvalues of graph $G$ in an interval $I$. In this paper, we show that if $G$ is not a cycle, then $m_G(n-g+3,n]\leq n-g$. Moreover, we…

For a graph $G$, let $L(G)$ and $Q(G)$ be the Laplacian and signless Laplacian matrices of $G$, respectively, and $\tau(G)$ be the number of spanning trees of $G$. We prove that if $G$ has an odd number of vertices and $\tau(G)$ is not…

Graph states provide a powerful framework for describing multipartite entanglement in quantum information science. In their standard formulation, graph states are generated by controlled-$Z$ interactions and naturally encode symmetric…

For a graph $G$, let $S_2(G)$ be the sum of the first two largest signless Laplacian eigenvalues of $G$, and $f(G)=e(G)+3-S_2(G)$. Very recently, Zhou, He and Shan proved that $K^+_{1,n-1}$ (the star graph with an additional edge) is the…

The $Q$-index of graph $G$ is the largest eigenvalue of the signless Laplacian matrix of $G$. Wang [Discrete Appl. Math. 356(2024)] proved the sharp upper bounds on the $Q$-index of leaf-free graphs with given size and characterized the…

Let $G$ be a graph of order $n$ and $Q_G(x)= det(xI-Q(G))= \sum_{i=1}^n (-1)^i \varphi_i x^{n-i}$ be the characteristic polynomial of the signless Laplacian matrix of a graph $G$. We give some transformations of $G$ which decrease all…

For a graph with $n$ vertices and $m$ edges, having Laplacian spectrum $\mu_1, \mu_2, \cdots,\mu_n$ and signless Laplacian spectrum $\mu^+_1,\mu^+_2, \cdots,\mu^+_n$, the Laplacian energy and signless Laplacian energy of $G$ are…

Let $G$ be a finite non abelian group. The centralizer graph of $G$ is a simple undirected graph $\Gamma_{cent}(G)$, whose vertices are the proper centralizers of $G$ and two vertices are adjacent if and only if their cardinalities are…

A vertex in a graph is said to be sedentary if a quantum state assigned on that vertex tends to stay on that vertex. Under mild conditions, we show that the direct product and join operations preserve vertex sedentariness. We also…