Abstract
An important property of strongly regular graphs is that the second subconstituent of any primitive strongly regular graph is always connected. Brouwer asked to what extent this statement can be generalized to distanceregular graphs. In this paper, we show that if γ is any vertex of a distanceregular graph Γ and t is the index where the standard sequence corresponding to the second largest eigenvalue of Γ changes sign, then the subgraph induced by the vertices at distance at least t from γ, is connected.
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1 Introduction
Let Γ be a distanceregular graph of diameter d. (For notation and definitions related to distanceregular graphs, see [4].) For primitive strongly regular graphs (the case d=2) it is known that Γ _{2}(γ), the subgraph of Γ induced by the vertices at distance 2 from a given vertex γ (also called the second subconstituent of γ), is connected. See [4, p. 86] or [3, Proposition 9.3.1] for an eigenvalue proof due to Haemers or [6] for a combinatorial proof of this fact. The properties of the second subconstituents of strongly regular graphs have been studied by many authors [3–6, 8] and their connectedness is a very important property of strongly regular graphs (cf. [3, Sect. 12.6]).
During the GAC5 conference [2] (see also [1]), Andries Brouwer asked whether this could be generalized to a statement that for general Γ and suitable t, the subgraph Γ _{≥t }(γ) is connected, where Γ _{≥t }(γ) is the subgraph of Γ induced by the vertices of distance at least t to γ. In this paper, we show that one can take for t the position where the standard sequence corresponding to the second largest eigenvalue changes sign.
2 Results
Let the distanceregular graph Γ have intersection array {b _{0},…,b _{ d−1};c _{1},…,c _{ d }} and put k=b _{0}, a _{ i }=k−b _{ i }−c _{ i } as usual. Define the tridiagonal matrices
and
Let ρ _{ i } be the largest eigenvalue of L _{ i }, and let σ _{ i } be the largest eigenvalue of M _{ i }. For γ∈V(Γ), Γ _{≤i }(γ) denotes the subgraph of Γ induced by the vertices of distance at most i from γ and Γ _{≥i }(γ) denotes the subgraph of Γ induced by the vertices at distance at least i from γ. Then ρ _{ i } is the largest eigenvalue of the subgraphs Γ _{≤i }(γ) of Γ, and σ _{ i } is the largest eigenvalue of the subgraphs Γ _{≥i }(γ) of Γ.
Proposition 1
(i) The ρ _{ i } are increasing: ρ _{ i }<ρ _{ j } when 0≤i<j≤d.
(ii) The σ _{ i } are decreasing: σ _{ i }>σ _{ j } when 0≤i<j≤d.
(iii) ρ _{ i }≤σ _{ d−i } for 0≤i≤d.
Proof
The Perron–Frobenius Theorem (see [4, Sect. 3.1], [7, Sect. 2.6] or [8, Sect. 8.8]) tells us that the largest eigenvalue of a graph is at least as large as the largest eigenvalue of a subgraph, and this gives (iii) because Γ _{≤i }(δ) is a subgraph of Γ _{≥d−i }(γ) when d(γ,δ)=d. That theorem moreover says that the inequality is strict for a proper subgraph of a connected graph, and this gives (i) because all Γ _{≤i }(δ) are connected, and also (ii) because each connected component of Γ _{≥j }(γ) is a proper subgraph of some connected component of Γ _{≥i }(γ). □
Let L=L _{ d }=M _{0}. Let Γ have the d+1 distinct eigenvalues k=θ _{0}>θ _{1}>⋯>θ _{ d }. Then the θ _{ i } are the eigenvalues of the matrix L. The standard sequence u=(u _{0},…,u _{ d })^{⊤} of an eigenvalue θ of Γ is the corresponding right eigenvector of L, so that Lu=θu, normalized such that u _{0}=1. From now on, let u denote the standard sequence of the second largest eigenvalue θ _{1} of Γ. It is known that it has precisely one sign change (see [4, Chap. 4]). In particular, u _{ d }<0.
Proposition 2
Let 1≤t≤d−1.

(i)
If u _{ t }=0, then ρ _{ t−1}=θ _{1}=σ _{ t+1}.

(ii)
If u _{ t }>0, then ρ _{ t−1}<θ _{1}<σ _{ t+1}.

(iii)
If u _{ t }<0, then ρ _{ t−1}>θ _{1}>σ _{ t+1}.’
Proof
Since the ρ _{ i } are increasing, and the σ _{ i } are decreasing, when proving (ii) we may assume that t is the largest index for which u _{ t }>0, and when proving (iii) we may assume that t is the smallest index for which u _{ t }<0.
Let x=(u _{0},…,u _{ t−1})^{⊤} and y=(u _{ t+1},…,u _{ d })^{⊤}. Now v:=L _{ t−1} x−θ _{1} x=(0,…,0,−b _{ t−1} u _{ t })^{⊤}, and the last component of v is zero, negative, resp. positive in the three cases of the proposition. Let z ^{⊤} be a positive left eigenvector of L _{ t−1} for its largest eigenvalue ρ _{ t−1}. We see that (ρ _{ t−1}−θ _{1})z ^{⊤} x=z ^{⊤} v is zero, negative, positive in the three cases, while z ^{⊤} x>0. This proves the lefthand inequalities.
Similarly, w:=M _{ t+1} y−θ _{1} y=(−c _{ t+1} u _{ t },0,…,0)^{⊤}, and the first component of w is zero, negative, resp. positive in the three cases of the proposition. Let z ^{⊤} be a positive left eigenvector of M _{ t+1} for its largest eigenvalue σ _{ t+1}. We see that (σ _{ t+1}−θ _{1})z ^{⊤} y=z ^{⊤} w is zero, negative, positive in the three cases, while z ^{⊤} y<0. This proves the righthand inequalities. □
Since ρ _{ i }≤σ _{ j } when i+j≤d, and ρ _{ i }<σ _{ j } when i+j<d, it follows that t>d/2 when u _{ t }<0, and t≥d/2 when u _{ t }≤0. See also [10] for some related results.
Theorem 3
Let Γ be a distanceregular graph of diameter d>0. Let u=(u _{0},…,u _{ d })^{⊤} be the standard sequence corresponding to the second largest eigenvalue of Γ. If u _{ t−1}>0, then Γ _{≥t }(γ) is connected for each vertex γ of Γ.
Proof
Let H:=Γ _{≥t }(γ), and let A _{ H } be its adjacency matrix. Let C be a connected component of H. Let z be a positive eigenvector of M _{ t } (indexed by {t,…,d}) for its largest eigenvalue σ _{ t }. The vector y defined by y _{ δ }=z _{ d(γ,δ)} for δ∈C, and y _{ δ }=0 for δ∈V(H)∖C is an eigenvector of H with eigenvalue σ _{ t }>θ _{1}. By eigenvalue interlacing (see [4, Sect. 3.3], [3, Sect. 2.5], [7, Chap. 5] or [8, Chap. 9]), H can have at most a single eigenvalue larger than θ _{1}, so H has only one connected component. □
3 Final remarks and open problems
The result stated in the introduction that the second subconstituent of a primitive strongly regular graph is connected, can be deduced from our work. The standard sequence u=(u _{0},u _{1},u _{2}) corresponding to the second largest eigenvalue θ _{1} of connected strongly regular graph has a sign change at u _{1} (as u _{0}=1,u _{1}=θ _{1}/k and u _{2} is negative) unless θ _{1}≤0. This happens if and only if the connected strongly regular graph is a complete multipartite graph.
We remark that there are many distanceregular graphs Γ of diameter d≥3 with the property that the subgraph induced by Γ _{ d }(γ) is not connected for any γ∈V(Γ). If Γ is a distanceregular graph with k _{ d }>1 and a _{ d }=0, then Γ _{ d }(γ) will be disconnected for any γ∈V(Γ). Distanceregular graphs having this property include antipodal rcovers with r≥3 and bipartite distanceregular graphs with k _{ d }≠1, but there are also primitive examples. The Patterson graph [4, p. 410] (for d=4), the Livingstone graph [4, p. 406] (for d=4) and the Biggs–Smith graph [4, p. 403] (for d=7) all have k _{ d }>1 and a _{ d }=0. The Coxeter graph [4, p. 382] has d=4, k _{ d }=6 and a _{ d }=1 (and thus Γ _{4}(γ) is isomorphic to 3K _{2}). The Sylvester graph [4, p. 394] has d=3, k _{ d }=10 and a _{ d }=1 (and thus Γ _{3}(γ) is isomorphic to 5K _{2}). The Odd graph Γ=O _{ d+1} [4, p. 259] is the graph whose vertex set is formed by all dsubsets of a set with 2d+1 elements, where two dsubsets are adjacent if and only if they are disjoint. This graph is distanceregular with degree d+1, diameter d and \(a_{d}=\lceil \frac{d+1}{2}\rceil\). For d≥3, one can check easily that the subgraph induced by Γ _{ d }(γ) is disconnected for any γ∈V(Γ).
If d∈{3,4}, then our results imply that the induced subgraph on Γ _{ d−1}(γ)∪Γ _{ d }(γ) is connected for any γ∈V(Γ), unless d=4 and Γ is an antipodal rcover for r≥3. All the known primitive distanceregular graphs that we have checked have Γ _{ d−1}(γ)∪Γ _{ d }(γ) connected. Is this statement true for all primitive distanceregular graphs? We showed it for diameter up to 4. If d=2s+1, then it is not clear when Γ _{≥s+1}(γ) is disconnected.
The Johnson graph J(n,d) has as vertices the subsets of order d of a set with n elements with two such subsets being adjacent if their intersection has size d−1. It is well known that J(n,d) is a distanceregular graph of diameter d and degree d(n−d) having b _{ j }=(d−j)(n−d−j) and c _{ j }=j ^{2} for each 0≤j≤d (see [4, Sect. 9.1]). For every γ∈V(J(n,d)), the induced subgraph Γ _{ d }(γ) is connected as is isomorphic to J(n−d,d). When n>d ^{2}, we can deduce that Γ _{ d }(γ) is connected using our previous results. The standard sequence corresponding to θ _{1}=(d−1)(n−d−1)−1=b _{1}−1 is \(u_{i}=1\frac{in}{d(nd)}\) for 0≤i≤d (see [4, Proposition 4.4.9]). It follows easily that if n divides d(n−d), then the standard sequence has a zero u _{ t }, where \(t=\frac{d(nd)}{n}\). If n does not divide d(n−d), then u _{ t−1}>0 and u _{ t }<0, when \(t=\lceil \frac{d(nd)}{n} \rceil\). Thus, when n>d ^{2}, then u _{ d−1}>0 and u _{ d }<0 which implies Γ _{ d }(γ) is connected for every γ∈V(J(n,d)). However, when n=4t−3, d=2t−2, we obtain u _{ t−1}>0 and u _{ t }<0. Also, when n=4t−1, d=2t−1, we obtain u _{ t−1}>0 and u _{ t }<0.
The Hamming graph H(d,q) has as vertex set the Cartesian product Q ^{d}, where Q is a set of q elements. Two vertices of H(d,q) are adjacent if they differ in exactly one position. It is well known that H(d,q) is a distanceregular graph of diameter d and degree d(q−1) having b _{ j }=(d−j)(q−1), c _{ j }=j for each 0≤j≤d (see [4, Sect. 9.2]). For every γ∈V(H(d,q)), the induced subgraph Γ _{ d }(γ) is connected as is isomorphic to H(d,q−1). When q>d, we can use our previous results to deduce that Γ _{ d }(γ) is connected. The standard sequence corresponding to θ _{1}=dq−d−q=b _{1}−1 is \(u_{i}=1\frac{iq}{d(q1)}\) for 0≤i≤d (see [4, Proposition 4.4.9]). It follows that if q divides d, then the standard sequence has a zero u _{ t }, where \(t=\frac{d(q1)}{q}\). If q does not divide d, then u _{ t−1}>0 and u _{ t }<0, when \(t=\lceil \frac{d(q1)}{q} \rceil\). Thus, when q>d, then u _{ d−1}>0 and u _{ d }<0 which implies Γ _{ d }(γ) is connected for every γ∈V(H(d,q)). However, when q=2, d=2t−1, then u _{ t−1}>0 and u _{ t }<0.
Motivated by the previous results, we call a distanceregular graph Γ of diameter d a generalized Shilla graph if the standard sequence corresponding to the second largest eigenvalue of Γ contains a zero. When d=3, this notion is the same as the notion of Shilla graph introduced by Koolen and Park in [9]. The previous examples show that many distanceregular graphs such as Hamming graphs or Johnson graphs are generalized Shilla graphs. It would be interesting to determine what other distanceregular graphs are generalized Shilla graphs. A related problem would be to classify all the distanceregular graphs of diameter d=3 such that Γ _{ d }(γ) is disconnected for some γ∈V(Γ). Such graphs would have to satisfy the condition u _{2}≤0. The case u _{2}=0 is particularly interesting and these graphs will have \(\theta_{1}=a_{3}=\frac{a_{1}+\sqrt{a_{1}^{2}+4k}}{2}\) (see Koolen and Park [9] for more details on such graphs). Another problem worth investigating is the relation between the index t, where the standard sequence corresponding to θ _{1} changes its sign, and the index s such that k _{ s }=max_{0≤i≤d } k _{ i }. If θ _{1}<k/2, then one can show that u _{ t−1}≥0 and u _{ t }<0 imply that b _{ t−2}≥k−θ _{1}>k/2 and c _{ t+1}>k−θ _{1}>k/2. This shows that b _{ t−3}>c _{ t−2} and c _{ t+2}>b _{ t+1} and hence k _{ t−3}<k _{ t−2} and k _{ t+1}>k _{ t+2}. Thus, when θ _{1}<k/2, the parameter s satisfies t−2≤s≤t+1. It would be interesting to determine how far apart can these parameters be in general.
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Acknowledgements
The authors are grateful to Andries Brouwer for many comments and suggestions that have greatly improved the original manuscript. Sebastian M. Cioabă thanks C.L. Toma for interesting conversations. This work was partially supported by a grant from the Simons Foundation (#209309 to Sebastian M. Cioabă). Jack H. Koolen was partially supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant number 20100028061). The authors thank the organizers (Edwin van Dam and Willem Haemers) and the participants of the GAC5 conference for creating a wonderful research atmosphere in which part of this work was done and the referees for useful comments and suggestions.
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Cioabă, S.M., Koolen, J.H. On the connectedness of the complement of a ball in distanceregular graphs. J Algebr Comb 38, 191–195 (2013). https://doi.org/10.1007/s1080101203985
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DOI: https://doi.org/10.1007/s1080101203985