Monotonic normalized heat diffusion for regular bipartite graphs with four eigenvalues

Let $X=(V, E)$ be a finite regular graph and $H_t(u, v), \, u, v \in V$, the heat kernel on $X$. We prove that, if the graph $X$ is bipartite and has four distinct Laplacian eigenvalues, the ratio $H_t(u, v)/H_t(u, u), \, u, v \in V,$ is monotonically non-decreasing as a function of $t$. The key to the proof is the fact that such a graph is an incidence graph of a symmetric 2-design.

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