Preface 1. Introduction 2. Graph operations and modifications 3. Spectrum and structure 4. Characterizations by spectra 5. Structure and one eigenvalue 6. Spectral techniques 7. Laplacians 8. Additional topics 9. Applications Appendix Bibliography Index of symbols Index.

An Introduction to the Theory of Graph Spectra: Spectral techniques

Periodicity of quantum walks defined by mixed paths and mixed cycles

Laplacian state transfer in edge complemented coronas

Twin vertices in simple unweighted graphs are vertices that have the same neighbours and, in the case of weighted graphs with possible loops, corresponding edges have equal weights. In this paper, we explore the role of twin vertices in quantum state transfer. In particular, we provide a characterization of periodicity, perfect state transfer, pretty good state transfer between twin vertices in a weighted graph with respect to its adjacency, Laplacian and signless Laplacian matrices. As an application, we characterize all simple unweighted double cones on regular graphs that exhibit periodicity, perfect state transfer, and pretty good state transfer.

pdf/quantum-state-transfer-between-twins-in-weighted-graphs-2bod2vwi.pdf

Quantum state transfer between twins in weighted graphs

Perfect state transfer on bi-Cayley graphs over abelian groups

Perfect state transfer in NEPS of complete graphs

Laplacian state transfer in Q-graph

Characterizing which graphs have Laplaican perfect state transfer or Laplacian pretty good state transfer is an interesting and desirable problem. In this paper, we investigate the existence of Lap...

Laplacian state transfer in edge coronas

Perfect state transfer in graphs is a concept arising from quantum physics and quantum computing. Given a graph $G$ with adjacency matrix $A_G$, the transition matrix of $G$ with respect to $A_G$ is defined as $H_{A_{G}}(t) = \exp(-\mathrm{i}tA_{G})$, $t \in \mathbb{R},\ \mathrm{i}=\sqrt{-1}$. We say that perfect state transfer from vertex $u$ to vertex $v$ occurs in $G$ at time $\tau$ if $u \ne v$ and the modulus of the $(u,v)$-entry of $H_{A_G}(\tau)$ is equal to $1$. If the moduli of all diagonal entries of $H_{A_G}(\tau)$ are equal to $1$ for some $\tau$, then $G$ is called periodic with period $\tau$. In this paper we give a few sufficient conditions for NEPS of complete graphs to be periodic or exhibit perfect state transfer.

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Perfect state transfer in NEPS of complete graphs

Laplacian state transfer in Q-graph

Laplacian state transfer in edge coronas

Perfect state transfer in NEPS of complete graphs