Emms, Hancock, Severini and Wilson's spectral characterization conjecture for strongly regular graphs

From papers

Let GG and HH be strongly regular graphs with the same set of parameters. For a graph FF, let U(F){\bf U}(F) denote its transition matrix, let U(F)3{\bf U}(F)^3 be its third power, let (U(F)3)+({\bf U}(F)^3)^+ denote its positive support, and let SpecSpec denote the spectrum. Emms, Hancock, Severini and Wilson's conjecture.

GHSpec((U(G)3)+)=Spec((U(H)3)+).G \cong H \quad\Longleftrightarrow\quad Spec\bigl(({\bf U}(G)^3)^+\bigr)=Spec\bigl(({\bf U}(H)^3)^+\bigr).

The conjecture proposes that, within a fixed parameter set of strongly regular graphs, the spectrum of the positive support of the cube of the transition matrix distinguishes isomorphic graphs. The supplied source does not state whether the conjecture has been resolved.

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Sources & referencesView supporting material

Primary source

Norio Konno and Iwao Sato, “On the relation between quantum walks and zeta functions”, arXiv:1103.0079 (2011).

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