Heat kernels on forms defined on a subgraph of a complete graph

The parametrix construction of the heat kernel on a graph \footnote{Keywords: heat kernels, graphs, parametrix, Bessel functions.}

“…Corollary 10 is particularly interesting since it gives a precise relation between the graph heat kernel on a graph G which is embedded in a domain Ω in terms of the geometric heat kernel associated to Ω, up to O(t 2 ) for small time t. Section 5 deals with derivation of expressions for the heat kernel in a general subgraph case. In Section 6 we show that our general result, applied to the setting of the complete graph on N vertices yields some of the main results from Section 3 of [LNY21]. We also give an alternative formulation for those results and explicitly compute the heat kernel on the graph obtained by removing a single edge from the complete graph.…”

“…Throughout this section we will employ some of the same considerations as in [LNY21], namely in the explicit computation of the graph convolution. In doing so, we will show the relationship between our Theorem 5 and Theorem 3.3 of [LNY21]. Let K N denote the complete graph K N on N vertices.…”

“…Let G be any subgraph of K N . Without loss of generality, we assume that G has N vertices, and take H G as a parametrix for the heat kernel H G on G. In order to compare our results with those from [LNY21], we will borrow and adapt some notation from [LNY21].…”

“…In the notation of formula (3.4) from [LNY21], In other words T F (x, y; t) = −(L 1 H G * F )(x, y; t).…”

“…A similar construction in the special case when G is embedded in a complete graph is considered in [LNY21]. In Section 6 below we relate our method to the approach in [LNY21] and show how our construction implies one of the main results of [LNY21].…”