I'm working on my senior capstone project for my physics major and have gotten caught up on some notation. The paper is about perfect state transfer on graphs with potential. The author places two conditions on the graphs for PST to be allowed; one of these conditions is as follows: $$\text{Every eigenvector } x \text{ of } H \text{ satisfies either } x(u)=x(v) \text{ or } x(u)=-x(v) $$ where $H$ is the graph hamiltonian (essentially just the adjacency matrix for the graph) and obviously $u$ and $v$ are vertices of the graph. The notation $x(u)$ was never clarified nor mentioned earlier in the paper. Is anybody familiar with this notation and willing to lend a hand? I will be happy to provide any additional information.
Spectral Graph Theory Notation
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graph-theory
spectral-graph-theory
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1I think you are looking at the vector space $K^V$, where $V$ is the vertex set and $K$ is the ground field (most likely $\mathbb{R}$ or $\mathbb{C}$). We can say that each vector $x$ is of the form $\big(x(v)\big)_{v\in V}$, i.e., $x(v)$ is the $v$-coordinate of $x$. This would make perfect sense with the adjacency matrix, as it is of the form $A=\big[A(u,v)\big]_{u,v\in V}$. – 2017-01-30
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0This makes fantastic sense. Thanks! – 2017-01-30