Abstract
The simplicial rook graph $$\mathrm{SR}(m,n)$$SR(m,n) is the graph of which the vertices are the sequences of nonnegative integers of length m summing to n, where two such sequences are adjacent when they differ in precisely two places. We show that $$\mathrm{SR}(m,n)$$SR(m,n) has integral eigenvalues, and smallest eigenvalue $$s = \max \left( -n, -{m \atopwithdelims ()2}\right) $$s=max-n,-m2, and that this graph has a large part of its spectrum in common with the Johnson graph $$J(m+n-1,n)$$J(m+n-1,n). We determine the automorphism group and several other properties.
Highlights
Let N be the set of nonnegative integers, and let m, n ∈ N
The simplicial rook graph SR(m, n) is the graph obtained by taking as vertices the vectors in Nm with coordinate sum n, and letting two vertices be adjacent when they differ in precisely two coordinate positions
For partitions of the set of coordinate positions {1, . . . , m} and integral vectors z indexed by that sum to n, let S,z be the set of all u ∈ X with i∈π ui = zπ for all π ∈
Summary
Let N be the set of nonnegative integers, and let m, n ∈ N. The graph SR(m, n) was studied in detail by Martin and Wagner [9] We settle their main conjecture and show that this graph has integral spectrum. The Johnson graph J (m, n) [with m n vertices, valency n(m − n) and eigenvalues (n − i)(m − n − i) − i with multiplicity m i. The graphs SR(m, n) and J (m + n − 1, n) both have m+n−1 n vertices and valency n(m − 1) These graphs resemble each other and have a large part of their spectrum in common. Proposition 10.1 The graph SR(m, n) is not determined by its spectrum when (a) m = 4 and n ≥ 3, or (b) n = 3 and m ≥ 4. We study the structure of the eigenspace for the eigenvalue −n
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