Eigenvalues and critical groups of Adinkras

The Critical Groups of Adinkras up to 2-Rank of Cayley Graphs

“…This could be done by considering the following matrix L(A) over Z[x]: fix an arbitrary color c, replace the diagonal entries of L(A) by x + (N − 1), and replace the off-diagonal entries ±1 corresponding to edges of color c by ±x. Since Z[x] is not a PID, it is not obvious that the SNF of L(A) exists, and it was conjectured in [17] that the SNF exists if and only if K(A) ∼ = (Z/(N 2 − N)Z) v/2 , which we now know the latter is true if and only if A is generic. It was only stated in [17] as a fact without proof that the forward direction is true, so we fill in the argument below: Proposition 4.1.…”

“…Since Z[x] is not a PID, it is not obvious that the SNF of L(A) exists, and it was conjectured in [17] that the SNF exists if and only if K(A) ∼ = (Z/(N 2 − N)Z) v/2 , which we now know the latter is true if and only if A is generic. It was only stated in [17] as a fact without proof that the forward direction is true, so we fill in the argument below: Proposition 4.1. When A is non-generic, the SNF of L(A) does not exist.…”

“…Our proof differs from the main strategy in [17], and uses monodromy pairing [19]. While the pairing is defined for any critical group, the regularity of Adinkras yields a simple description of it [15].…”

“…Therefore, Adinkras can be thought of as a signed analogue of Cayley graphs for supersymmetry algebras. In [17], the authors initiated the problem, and proved the following using a "deform to Z[x]-modules" argument:…”

Critical groups of strongly regular graphs and their generalizations

“…These signed graphs generalize regular two-graphs, and are of special interest recently for their applications in the proof of the sensitivity conjecture [14] and construction of line systems in Euclidean space [25]. Moreover, a family of decorated graphs known as Adinkras, introduced by physicists to encode special supersymmetry algebras and Clifford algebras (or representations thereof) [9,15], can be shown to have exactly two distinct Laplacian eigenvalues as well [16].…”