Cospectral mates for generalized Johnson and Grassmann graphs

We provide three infinite families of graphs in the Johnson and Grassmann schemes that are not uniquely determined by their spectrum. We do so by constructing graphs that are cospectral but non-isomorphic to these graphs.


Introduction
A major problem in the field of spectral graph theory is to decide whether a given graph is determined by its spectrum (the eigenvalues of its adjacency matrix).This problem has been solved for many families of graphs; sometimes by proving that the spectrum determines the graph, and sometimes by constructing cospectral mates (non-isomorphic graphs with the same spectrum).Although a conjecture by the third author states that almost all graphs are determined by their spectrum, many well-known graphs seem to have cospectral mates.This is, in particular, the case for several graphs in the Johnson and Grassmann association schemes, see for instance [2,3,4,6,9].We contribute to this line of research by constructing cospectral mates for a number of open cases in the Johnson and Grassmann schemes.For this, we use switching techniques due to Godsil and McKay [5] and Wang, Qiu and Hu [11].
This paper is structured as follows.In Section 2, we introduce the necessary definitions.Section 3 contains an overview of previous results on the cospectrality of generalized Johnson and Grassmann graphs.In Section 4, we prove that the generalized Johnson graph J {2} (n, 4), n ≥ 8, is not determined by its spectrum, thereby solving the second open problem of [2].In Section 5, we show that the generalized Johnson graph J {1,2,... k−1 2 } (2k, k) with k ≥ 5, k odd, is not determined by its spectrum.In Section 6, we prove that the q-Kneser graph K q (n, k) is not determined by its spectrum if q = 2.In Section 7 we provide three specific generalized Johnson graphs that are not determined by their spectrum and pose some open questions.Finally, the Appendix contains an overview of all cospectrality results for small graphs in the aforementioned schemes.

Preliminaries
Let Γ = (V, E) denote a simple loopless graph.The spectrum of Γ is the multiset of eigenvalues of the adjacency matrix of Γ. Graphs are cospectral if they have the same spectrum.Two graphs that are cospectral but not isomorphic, are called cospectral mates.
A graph is determined by its spectrum (DS) if it has no cospectral mate.Otherwise, it is not determined by its spectrum (NDS).
One of the most well-known tools for constructing cospectral graphs was introduced by Godsil and McKay [5].
Theorem 1 (GM-switching [5]).Let Γ be a graph and {C 1 , C 2 , . . .C k , D} a partition of V (Γ) such that the following hold for all i, j ∈ {1, 2, . . ., k}: For all i ∈ {1, 2, . . ., k}, every u ∈ C i and every v ∈ D that has exactly 1  2 |C i | neighbours in C i , reverse the adjacency between u and v.The resulting graph is cospectral with Γ.We say that it is obtained from Γ by GM-switching with respect to C 1 , C 2 , . . .C k .
Another switching technique to construct cospectral graphs was recently introduced by Wang, Qui and Hu [11].
Theorem 2 (WQH-switching [11]).Let Γ be a graph and C 1 , C 2 disjoint subsets of V (Γ) such that the following hold: (ii) There exists a constant c such that for all i, j ∈ {1, 2}, i = j and every vertex of C i , the number of neighbours in C i minus the number of neighbours in C j is equal to c.
(c) the same number of neighbours in C 1 and C 2 .
For every u ∈ C 1 ∪ C 2 and every v / ∈ C 1 ∪ C 2 for which (a) or (b) holds, reverse the adjacency between u and v.The resulting graph is cospectral with Γ.We say that it is obtained from Γ by WQH-switching with respect to C 1 and C 2 .
We should mention that a more general version of WQH-switching was introduced in [10].
Let n, k be positive integers with k ≤ n.We now define the graphs of our interest, namely those in the Johnson and Grassmann schemes.Definition 3. Let S ⊆ {0, 1, . . ., k − 1}.The generalized Johnson graph J S (n, k) has as vertices the k-subsets of {1, . . ., n}, where two vertices are adjacent if their intersection size is in S.
In particular, J {0} (n, k) is the Kneser graph K(n, k) and J {k−1} (n, k) is the Johnson graph J(n, k).
If we replace sets by subspaces and sizes by dimensions in Definition 3, we obtain the definition of a generalized Grassmann graph.A vector space of dimension k is called a k-space for short.Furthermore, let F q denote the finite field of order q (q is a prime power).Definition 4. Let S ⊆ {0, 1, . . ., k − 1}.The generalized Grassmann graph J q,S (n, k) has as vertices the k-subspaces of F n q , where two vertices are adjacent if their intersection dimension is in S.
In particular, J q,{0} (n, k) is the q-Kneser graph K q (n, k) and J q,{k−1} (n, k) is the Grassmann graph J q (n, k).
From now on, for all the aforementioned graphs, we assume that k ≤ n/2, since J S (n, k) is isomorphic to J {s+n−2k|s∈S} (n, n − k).We also assume that |S| ≤ k/2 because the complement of J S (n, k) is J {1,...,n}\S (n, k), and a regular graph is DS if and only if its complement is DS.Note that these observations also hold for Grassmann graphs.Also note that if S = ∅, the graph is complete or edgeless and therefore trivially DS.Therefore, we also assume that |S| ≥ 1, and in particular, k ≥ 2.

Previous results
The following results about the cospectrality of generalized Johnson and Grassmann graphs have been obtained by various authors.We present them in a chronological order.
Let q be a prime power and 1 ≤ k ≤ n/2.
Theorem 11 The authors of [2] found, computationally, that the graph J {2} (8, 4) has a cospectral mate by GM-switching with respect to switching sets of size 8, which implies that it is NDS.
So there are n−6 2 such vertices.
(iii) If a vertex is adjacent to v and w but does not contain 1 or 2, it must be equal to {3, 4, 5, 6}.
We can now conclude that v and w have n−6 Theorem 13.J {2} (n, 4) is not determined by its spectrum if n ≥ 8.
the number of neighbours of v in C 1 and C 2 only depend on its intersection size with {1, 2, 3} and {4, 5, 6}.
We are left to prove that the graph obtained by WQH-switching with respect to C 1 and C 2 , is not isomorphic to the original one.
Consider the vertices v = {1, 2, 3, 4} ∈ C 1 and w = {1, 4, 5, 7} / ∈ C 1 ∪ C 2 .The neighbours of w are preserved by switching, since |w ∩ {1, 2, 3}| = 1 and |w ∩ {4, 5, 6}| = 2 (see Table 1).In particular, v and w are adjacent in both Γ and Γ ′ .We show that they have more common neighbours in Γ ′ than in Γ.For this, we only need to consider neighbours of w for which the adjacency with v is changed during the switching process.We observe from Table 1 that the switching only affects those vertices which have 2 elements in one of the sets {1, 2, 3}, {4, 5, 6} and none in the other.Neighbours of w that contain 2 elements of {1, 2, 3} and none of {4, 5, 6}, are of the form {1, 2, 7, a} or {1, 3, 7, a} with a ≥ 8.So there are a total of 2(n − 7) vertices that are adjacent to v and w in Γ but not adjacent to v in Γ ′ .On the other hand, neighbours of w that contain no element of {1, 2, 3} and 2 elements of {4, 5, 6}, are of the form {4, 5, a, b}, {4, 6, 7, a} or {5, 6, 7, a} with a, b ≥ 8.They are not adjacent to v in Γ, but become adjacent to v after switching.So n−7 2 + 2(n − 7) new common neighbours are created.We conclude that v and w have n−7 2 more common neighbours in Γ ′ than any two adjacent vertices in Γ, which is a strictly positive difference, except when n = 8.In order to solve this one specific case, we take a look at an other pair of vertices.
We proved that there are always two neighbours in Γ ′ which have strictly more common neighbours than any two neighbours in Γ.So Γ and Γ ′ cannot be isomorphic.
The next result uses GM-switching with respect to two switching sets.Our argument is inspired by the proof of Theorem 10(i), as J {1,2,... k−1 2 } (2k, k) can be seen as two copies of J {0,1,... k−3 2 } (2k − 1, k − 1), connected in a certain way.While our first switching set corresponds to the one used to show Theorem 10(i), the second one is given by its complement.
There is a bijection from C 1 to C 2 , given by the complement.We first check the conditions of Theorem 1 in order to show that C 1 and C 2 form a GM-switching set of Γ.It is immediate that , then it is adjacent with no vertices of C 1 and all vertices of C 2 .We conclude that C 1 and C 2 fulfil the GM-switching conditions.
Next, we show that the graph Γ ′ obtained from Γ by GM-switching with respect to C 1 and C 2 ), is not isomorphic to Γ. Define v := [k] ∈ C 1 and w := {2, 3, . . ., k + 1} / ∈ C 1 ∪ C 2 .We claim that v and w have strictly less common neighbours after switching.For this, it suffices to count the number of common neighbours that are lost or added during the GM-switching process.Therefore, we only consider those neighbours of w that have . First consider the common neighbours that are added.They should not be adjacent to v in the original graph, which means that they must contain the element k.Adjacency with w implies that they must also contain the element 1 and not the element k + 1.Thus, there are one of which is k added common neighbours of v and w.Now consider the common neighbours that are lost.Since these vertices must originally be adjacent to v, they may not contain k.If they contain the element 1, then they will automatically be adjacent to w.So there are at least common neighbours of v and w that are lost during switching, which is already strictly more than the number of added common neighbours.Now, assume by contradiction that Γ ∼ = Γ ′ .Let λ(u, w) denote the number of common neighbours of u and w, for any vertex u and w defined as before.Since Γ is vertextransitive, the multiset Λ(w) = {λ(u, w) | u ∼ w} remains the same after switching.The value of λ(u, w) stays the same if u / ∈ C 1 ∪ C 2 , so we can restrict ourselves to vertices in C 1 ∪ C 2 , i.e. the multiset is invariant.We now apply the same reasoning as in [2].Define The automorphism induced by the permutation (k Together with the observation that λ(v, w) = λ(v ′ , w), we see that λ(v, w) is the same for Γ and Γ ′ , a contradiction with the above.We conclude that Γ and Γ ′ are cospectral mates.Thus, Γ is NDS.
6 q-Kneser graphs are NDS if q = 2 In this section, we prove that the q-Kneser graph K 2 (n, k) = J 2,{0} (n, k) is NDS (under the assumption that k ≥ 2, otherwise the graph is trivially DS) using GM-switching with respect to a switching set of size 4.This result can be seen as a q-analog of Theorem 9(i) in the case that q = 2 (and without the extra condition).
Theorem 15.K 2 (n, k) is not determined by its spectrum.
Let Γ ′ be the graph obtained from Γ by GM-switching with respect to C. We prove that Γ and Γ ′ are non-isomorphic.Let τ be a (k −1)-space that intersects p 1 p 2 p 3 π trivially and consider the pairwise non-adjacent k-spaces p 1 τ , p 2 τ and p 4 τ in Γ ′ .Then there is exactly one k-space that is adjacent to p 1 τ but not adjacent to both p 2 τ and p 4 τ in Γ ′ .This is the k-space p 1 p 3 π.Indeed, a k-space outside C that intersects both p 2 τ and p 4 τ , must intersect p 1 τ as well.So the only spaces that can meet this property are in C. Since the switching reverses adjacency for p 1 τ , p 2 τ and p 4 τ , we get that p 1 p 3 π is adjacent to p 1 τ and not adjacent to p 2 τ and p 4 τ , while the other elements of C are not.
Now consider three arbitrarily chosen non-adjacent k-spaces in Γ.Call them α, β and γ.Non-adjacency here means that they intersect one another.We prove that the number of k-spaces that intersects α trivially but intersects both β and γ non-trivially, is never equal to one.Let δ be such a space (if it does not exist, we are done).Choose two 1-spaces (projective points) p ∈ β ∩ δ and q ∈ γ ∩ δ.The 2-space (projective line) pq does not intersect α, as it lies in δ.There are 2 k(k−2) n−k−1 k−2 2 > 1 different k-spaces through pq that intersect α trivially, and they all meet the above property.Since the number of k-spaces with this property is different for Γ and Γ ′ , while it should be invariant under isomorphism, we conclude that Γ and Γ ′ are not isomorphic.

Open problems
Besides the three infinite families of graphs that are NDS, we also found three "sporadic" generalized Johnson graphs with a cospectral mate.For each of these three graphs, we provide a pair of switching sets that produces a cospectral mate, see the table below.The fact that they are non-isomorphic was checked by computer.graph method switching sets J {1} (11,4) WQH It would be nice to see if any of these switching sets can be extended to an infinite family of generalized Johnson graphs that are NDS.We believe that J {2,4,... } (2k, k) is NDS if k ≥ 4, following a similar argument as in [6], but with two switching sets instead of one.
More generally, it remains an open problem to determine the cospectrality of other graphs in the Johnson and Grassmann schemes (entries "?" in the tables in the Appendix).The smallest open case is K(9, 3), see also [2,Section 5].

Figure 1 :
Figure 1: Elements of C 2 include [k − 1] and one other element.