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We study Aldous' conjecture that the spectral gap of the interchange processon a weighted undirected graph equals the spectral gap of the random walk onthis graph. We present a conjecture in the form of an inequality, and provethat this inequality implies Aldous' conjecture by combining an interlacingresult for Laplacians of random walks on weighted graphs with representationtheory. We prove the conjectured inequality for several important instances. Asan application of the developed theory, we prove Aldous' conjecture for a largeclass of weighted graphs, which includes all wheel graphs, all graphs with fourvertices, certain nonplanar graphs, certain graphs with several weighted cyclesof arbitrary length, as well as all trees.  Caputo, Liggett, and Richthammer have recently resolved Aldous' conjecture,after independently and simultaneously discovering the key ideas developed inthe present paper.

Interlacings for random walks on weighted graphs and the interchange process