Isomorphic and Nonisomorphic, Isospectral Circulant Graphs

Constructing cospectral signed graphs

A well--known fact in Spectral Graph Theory is the existence of pairs of isospectral nonisomorphic graphs (known as PINGS). The work of A.J. Schwenk (in 1973) and of C. Godsil and B. McKay (in 1982) shed some light on the explanation of the presence of isospectral graphs, and they gave routines to construct PINGS. Here, we consider the Godsil-McKay--type routines developed for graphs, whose adjacency matrices are $(0,1)$-matrices, to the level of signed graphs, whose adjacency matrices allow the presence of $-1$'s. We show that, with suitable adaption, such routines can be successfully ported …

关于等谱整数循环图

On Isospectral Integral Circulant Graphs

本文研究整数循环图的等谱同构问题，证明了在多数素数取值下及特定形式（如p^2q^n和p^2q^2）满足整数谱Adàm性质，推进了该开放问题的解决。

Graph isomorphism and Gaussian boson sampling

We introduce a connection between a near-term quantum computing device, specifically a Gaussian boson sampler, and the graph isomorphism problem. We propose a scheme where graphs are encoded into quantum states of light, whose properties are then probed with photon-number-resolving detectors. We prove that the probabilities of different photon-detection events in this setup can be combined to give a complete set of graph invariants. Two graphs are isomorphic if and only if their detection probabilities are equivalent. We present additional ways that the measurement probabilities can be combine…