Binary Representations of Regular Graphs

For any 2-distance set X in the n-dimensional binary Hamming spaceHn, let ΓX be the graph with X as the vertex set and with two vertices adjacent if and only if the distance between them is the smaller of the two nonzero distances in X. The binary spherical representation number of a graph Γ, or bsr Γ , is the least n such that Γ is isomorphic to ΓX , where X is a 2-distance set lying on a sphere in Hn. It is shown that if Γ is a connected regular graph, then bsr Γ ≥ b − m, where b is the order of Γ and m is the multiplicity of the least eigenvalue of Γ, and the case of equality is characterized. In particular, if Γ is a connected strongly regular graph, then bsr Γ b − m if and only if Γ is the block graph of a quasisymmetric 2-design. It is also shown that if a connected regular graph is cospectral with a line graph and has the same binary spherical representation number as this line graph, then it is a line graph.


Introduction
The subject of this paper is mutual relations between regular and strongly regular graphs, 2-distance sets in binary Hamming spaces, and quasisymmetric 1-and 2-designs.
The following relation between strongly regular graphs and 2-distance sets in Euclidean spaces is well known cf. 1, Theorem 2.23 : if m is the multiplicity of the least eigenvalue of a connected strongly regular graph Γ of order n, then the vertex set of Γ can be represented as a set of points, lying on a sphere in R n−m−1 , so that there exist positive real numbers h 1 < h 2 such that the distance between any two distinct vertices is equal to h 1 if they are adjacent as vertices of Γ and it is equal to h 2 otherwise.This result was recently generalized to all connected regular graphs in 2 .It has also been proved in 2 that, given n and m, such a representation of a connected regular graph in R n−m−2 is not possible.
The notion of a 2-distance set representing a graph makes sense for any metric space, and the spaces of choice in this paper are the binary Hamming spaces.We will show Theorem 3.3 that the dimension of a binary Hamming space, in which a connected regular

Preliminaries
All graphs in this paper are finite and simple, and all incidence structures are without repeated blocks.For a graph Γ, |Γ| denotes the order of Γ, that is, the number of vertices.If x and y are vertices of a graph Γ, then x ∼ y means that x and y are adjacent, while x / ∼y means that x and y are distinct and nonadjacent.Two graphs are said to be cospectral if their adjacency matrices have the same characteristic polynomial.
Throughout the paper we use I to denote identity matrices and J to denote square matrices with every entry equal to 1.The order of I and J will be always apparent from the context.We denote as 0 and 1 vectors columns, rows, points with all entries coordinates equal to 0 or all equal to 1, respectively.In examples throughout the paper we will use digits and letters to denote elements of a small set and omit braces and commas when a subset of such a set is presented; for example, we will write 1350b instead of {1, 3, 5, 0, b}.
Definition 2.1.The binary Hamming space H n consists of all n-tuples a a 1 , a 2 , . . ., a n with each a i equal to 0 or 1.When it is convenient, one identifies a with the set {i ∈ n : a i 1}.Remark 2.2.The sphere of radius k in H n , centered at a, coincides as a set with the sphere of radius n − k centered at the opposite point b 1 − a.This allows us to assume, when needed, that the radius of a sphere does not exceed n/2.A sphere of radius k in H n centered at 0, regarded as a subset of R n , is the intersection of the unit cube and the hyperplane International Journal of Combinatorics 3 Remark 2.3.For n ≥ 2, the distance between any two points of a spherical set in H n is even.Definition 2.4.An incidence structure without repeated blocks is a pair D V, B , where V is a nonempty finite set of points and B is a nonempty set of subsets of V blocks .The cardinality of the intersection of two distinct blocks is called an intersection number of D. An incidence structure is said to be quasisymmetric if it has exactly two distinct intersection numbers.For a nonnegative integer t, an incidence structure D is called a t-design if all blocks of D have the same cardinality and every set of t points is contained in the same number of blocks.A t-design D with an points versus blocks incidence matrix N is called nonsquare if N is not a square matrix, and it is called nonsingular if det NN / 0. A 2-design is also called a v, b, r, k, λ -design, where v is the number of points, b is the number of blocks, r is the replication number, that is, the number of blocks containing any given point, k is the block size, and λ is the number of blocks containing any given pair of points.
With any quasisymmetric incidence structure we associate its block graph.
where N is an incidence matrix of D and A is an adjacency matrix of the block graph of D. If D is a v, b, r, k, λ -design, then NN r − λ I λJ.Therefore, det NN rk r − λ v−1 / 0, so D is nonsingular.For these and other basic results on designs and regular graphs, see 1 or 4 .Definition 2.7.Let X {x 1 , x 2 , . . ., x b } be a 2-distance set of cardinality b in H n , and let h 1 < h 2 be the nonzero distances in X.One denotes as Γ X the graph whose vertex set is X and the edge set is the set of all pairs {x i , x j } with h x i , x j h 1 .For i 1, 2, . . ., b, let x i x i1 , x i2 , . . ., x in and B i {j ∈ n : x ij 1}, so x i is the characteristic vector of B i .Let B {B 1 , B 2 , . . ., B b }.One denotes as D X the incidence structure n , B .Remark 2.8.If X is a spherical 2-distance set centered at 0, then the incidence structure D X is a quasisymmetric 0-design and Γ X is its block graph.Proposition 2.9.Let X be a 2-distance set in H n , and let Proof.Suppose h 2 > 2h 1 .If x, y, and z are distinct vertices of Γ X such that x ∼ y and x ∼ z, then the triangle inequality implies that y ∼ z.Therefore, all neighbors of x form a connected component of Γ X .Since Γ X is not a complete graph, it is not connected; a contradiction.x i1 , x i2 , . . ., x i,e bk ∈ H e bk , where x ij 1 if and only if j ∈ X i ∪ Y i .Then, for 1 ≤ i < j ≤ b, the distance between points x i and x j is equal to 2 k − 2 if the ith and jth vertices of Γ are adjacent, and it is equal to 2k otherwise.Since Γ is not a complete graph, the set {x 1 , x 2 , . . ., x b } is a 2-distance set representing Γ in H e bk , and this set lies on a sphere of radius k centered at 0.
If the graph Γ in the above proof is regular, we do not need to add columns to its incidence matrix N. Proposition 2.12.If Γ is a noncomplete regular graph with e ≥ 1 edges, then bsr Γ ≤ e. Theorem 5.1 implies that if Γ is a cycle, then its binary spherical representation number equals the number of edges.
For any graph G, the line graph of G, denoted as L G , is the graph whose vertex set is the edge set of G; two distinct vertices of L G are adjacent if and only if the corresponding edges of G share a vertex.Line graphs are precisely the graphs representable by spherical 2-distance sets of radius 2.

Proposition 2.13. A graph Γ can be represented in H n by a spherical 2-distance sets of radius 2 if and only if Γ is isomorphic to the line graph of a graph of order n.
Proof.If Γ L G , where G is a graph of order n, then the columns of an incidence matrix of G form a 2-distance subset of H n of radius 2 representing Γ.Conversely, let X be a 2-distance subset of H n of radius 2 centered at 0 and representing a graph Γ.Let G be a graph whose incidence matrix coincides with an incidence matrix of D X .Then |G| n and Γ is isomorphic to L G .Remark 2.14.Let G be a regular graph of degree r, and let X be the set of columns of an incidence matrix N of G. Then D X is a quasisymmetric 1-design with block size 2 and replication number r and N is its incidence matrix.If r ≥ 3, this design is non-square.The next result Proposition 2.3 in 5 yields a necessary and sufficient condition for this 1-design to be nonsingular.

Proposition 2.15. If N is an incidence matrix of a graph Γ of order n and c is the number of connected components of Γ, then
2.1

Lower Bounds
The main tool in obtaining a lower bound on bsr Γ is the following classical theorem of distance geometry.
If X is a finite set in R n , then, for any z ∈ R n , the Schoenberg matrix S z X is positive semidefinite and rank S z X ≤ n.
We will now derive a sharp lower bound on the binary spherical representation number of a connected regular graph.Proof.Let bsr Γ n, and let Γ be isomorphic to Γ X , where X is a spherical 2-subset of H n .Let h 1 < h 2 be the nonzero distances in X and k the radius of a sphere in H n containing X. Without loss of generality, we assume that this sphere is centered at 0. Then z k/n, k/n, . . ., k/n is the center of an Euclidean sphere containing X.The radius of this sphere is equal to k n − k /n.Let A be an adjacency matrix of Γ X .Then the matrix is the Schoenberg matrix of the set X with respect to z.
Let d be the degree of Γ, ρ 0 d > ρ 1 > • • • > ρ s ρ all distinct eigenvalues of A, and m 0 1, m 1 , . . ., m s m their respective multiplicities.Then the eigenvalues of S are with 1 as an eigenvector, and, for i 1, 2, . . ., s, with eigenvectors orthogonal to 1 .For 0 ≤ i ≤ s, the multiplicity of σ i is at least m i .It is greater than m i if σ i σ 0 , i / 0. Theorem 3.2 implies that all eigenvalues of S are nonnegative, so On the other hand, since both X and z lie in the hyperplane Suppose now that n |Γ| − m.Then rank S |Γ| − m − 1, and therefore σ s σ 0 0. From σ 0 0 we derive

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The incidence structure D X n , B has n points, |Γ| blocks, all of cardinality k, and two intersection numbers, For each i ∈ n , let r i denote the number of blocks of D X containing i. Fix a block C and count in two ways pairs B, i , where B ∈ B, B / C, and i ∈ C ∩ B: Using this equation and 3.6 , we derive Since each i ∈ n contributes r 2 i into the left-hand side of this equation, we obtain that

3.10
On the other hand, counting in two ways pairs i, B with B ∈ B and i ∈ B yields Therefore, r i r |Γ|k/n for i 1, 2, . . ., n.Thus, D X is a quasisymmetric 1-design.Note that we have derived this result from 3.5 rather than from a stronger equation rank S |Γ| − m − 1.Since n < |Γ|, the 1-design D X is non-square, so we have to show that it is nonsingular.The incidence matrix N of D X satisfies the equation

3.13
International Journal of Combinatorics 7 Therefore, the eigenvalues of N N are and since rank N N ≤ n, we obtain that τ s 0 and τ i > 0 for 0 ≤ i ≤ s−1.Since the multiplicity of τ s is the same as the multiplicity of ρ s , we have rank N N n.Therefore, rank NN n, and then det NN / 0, that is, D X is nonsingular.Suppose now that Γ is the block graph of a nonsquare nonsingular quasisymmetric 1-design D with intersection numbers α < β.The design D has less points than blocks, so let b be the number of blocks and b − m the number of points.We have to show that m is the multiplicity of the least eigenvalue of Γ and that bsr Γ b − m.Let N be an incidence matrix of D and X the set of all columns of N regarded as points in H b−m .Then X is a 2-distance set and D is D X .The set X lies on a sphere of radius k centered at 0, where k is the cardinality of each block of D, and the nonzero distances in X are h 1 2 k − β and h 2 2 k − α .
Matrix N satisfies 3.13 with A being an adjacency matrix of Γ.Let ρ 0 > ρ 1 > • • • > ρ s be all distinct eigenvalues of A. Then the eigenvalues of N N are given by 3.14 .Since N has more rows than columns, we have τ s 0. Since det NN / 0, the sum of the multiplicities of the nonzero eigenvalues of N N is b − m, so the multiplicity of τ s is equal to m.Therefore, the multiplicity of ρ s is equal to m, and then bsr It has been shown in the course of this proof that if bsr Γ |Γ| − m, then σ 0 0, which implies 3.5 , and σ s 0, which implies If h 2 /h 1 ρ/ ρ 1 , then ρ is rational, so ii implies that the following useful result.
An infinite family of regular graphs attaining the lower bound of Theorem 3.3 is given in the following example.

Strongly Regular Graphs
For strongly regular graphs we first obtain a sharp upper bound for the binary spherical representation number.
Proposition 4.1.If Γ is a connected strongly regular graph of order n, then bsr Γ ≤ n.
Proof.Let Γ be an srg n, d, λ, μ , and let A be an adjacency matrix of Γ.
Let X be the set of rows of A I regarded as points in H n .Then the distance between two distinct points of X is equal to 2 d − λ − 1 if the points correspond to adjacent vertices of Γ; otherwise, it is equal to 2 d − μ 1 .Thus, X is a 2-distance set in H n , lying on a sphere of radius d 1 centered at 0, and, if Example 4.4.Let Γ be the complement of the cycle C 7 .The least eigenvalue of Γ is irrational, so bsr Γ ≥ 7. Suppose bsr Γ 7, and let Γ be isomorphic to Γ X , where X is a 2-subset of H 7 with nonzero distances h 1 < h 2 , lying on a sphere of radius k centered at 0. Since h 1 and h 2 are even, h 2 ≤ 7 and h 2 ≤ 2h 1 Proposition 2.9 , we have h 1 2, h 2 4 or h 1 4, h 2 6.In either case, Theorem 3.5 iii yields an equation without integer solutions.Thus, bsr Γ ≥ 8, so the strong regularity in Proposition 4.1 is essential.
Remark 4.5.There are 167 nonisomorphic strongly regular graphs with parameters 64, 18, 2, 6 8 .The least eigenvalue of these graphs is −6 of multiplicity 18. Theorem 3.3 and Proposition 4.1 imply that if Γ is any of these 167 graphs, then 46 ≤ bsr Γ ≤ 64.Therefore, there are nonisomorphic graphs with these parameters having the same binary spherical representation number.
Also, there are 41 nonisomorphic strongly regular graphs with parameters 29, 14, 6, 7 8 .The least eigenvalue of these graphs is irrational, so by Corollary 4.2 the binary spherical representation number of all these graphs is 29.Theorem 3.3 for regular graphs can be rectified if the graph is strongly regular.
Let A be an adjacency matrix of Γ. Theorem 3.4 ii implies that

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Let N be an incidence matrix of D X .Then

4.2
From these equations we determine the eigenvalues of N N: τ 0 kr, τ 1 k − α β − α ρ 1 r − λ, and τ 2 k − α β − α ρ 2 0. Their respective multiplicities are 1, b − m − 1, and m.Therefore, the eigenvalues of NN are τ 0 of multiplicity 1 and τ 1 of multiplicity b−m−1.Since NN J krJ, the eigenspace E 0 of NN corresponding to the eigenvalue τ 0 is generated by 1.Therefore, E 1 E ⊥ 0 is the eigenspace corresponding to the eigenvalue τ 1 .On the other hand, the matrix M r − λ I λJ has the same eigenvalues with the same respective eigenspaces.Thus, NN M, and therefore D X is a quasisymmetric 2-design with intersection numbers α and β.The graph Γ is the block graph of this design.
Example 4.7.The Cocktail Party graph CP n has 2n vertices split into n pairs with two vertices adjacent if and only if they are not in the same pair.It is the block graph of a quasisymmetric 2-design if and only if the design is a Hadamard 3-design with n 1 points cf.
4, Theorem 8.2.23 .The least eigenvalue of CP n is −2 of multiplicity n − 1.By Theorem 4.6, bsr CP n ≥ n 1 and bsr CP n n 1 if and only there exists a Hadamard matrix of order n 1.This example shows that it is hard to expect a simple general method for computing the binary spherical representation number of a strongly regular graph.

Line Graphs
In this section we determine the binary spherical representation number for the line graphs of regular graphs.If N is an incidence matrix of a graph G, then N N 2I A, where A is an adjacency matrix of the line graph Γ L G .Let G be connected and have n vertices and e edges.If e > n, then the least eigenvalue of N N is 0, and therefore the least eigenvalue of Γ is ρ −2.Since matrices NN and N N have the same positive eigenvalues, Proposition 2.15 implies that the multiplicity of ρ is equal to e − n if the graph G is not bipartite, and it is equal to e − n 1 if G is a connected bipartite graph.If e n, then G is a cycle, so Γ C n is a cycle of order n too.If n is even, then the least eigenvalue of C n is −2 of multiplicity 1; if n ≥ 5 is odd, then the least eigenvalue of C n is irrational.See 9 for details.Proof.Let Γ be the line graph of a connected regular graph G of order n ≥ 4 and degree d.
Then Γ is a connected regular graph of order nd/2 and degree 2d − 2. The columns of an incidence matrix of G form a spherical 2-distance set in H n representing Γ, so bsr Γ ≤ n.Suppose first that d 2, that is, G is C n , and that n is odd.Then the least eigenvalue of Γ is irrational.Therefore, bsr Γ ≥ n by Corollary 3.6, so bsr Γ n.Suppose now that d ≥ 3 and the graph G is not bipartite.From Proposition 2.15, the multiplicity of the least eigenvalue of Γ is nd/2−n, and then Theorem 3.3 implies that bsr Γ ≥ n, so bsr Γ n.Suppose finally that G is a bipartite graph this includes the case G C n with even n .Then the least eigenvalue of Γ is −2 and its multiplicity is nd/2 − n 1.Therefore, by Theorem 3.3, bsr Γ ≥ n − 1. Suppose bsr Γ n − 1. Theorem 3.4 ii implies that h 2 2h 1 and then the condition i of Theorem 3.4 can be rewritten as If n is odd, then n − 1 n − 2 and n are relatively prime; if n is even, then n − 1 n − 2 /2 and n/2 are relatively prime.In either case, n The graph L 2 n is the line graph of the bipartite graph with the bipartition sets of cardinality n.The following corollary generalizes the well-known result 10 that these graphs are not block graphs of quasisymmetric 2-designs.
Corollary 5.2.The line graph of a connected regular graph G with more than three vertices is the block graph of a nonsquare nonsingular quasisymmetric 1-design if and only if G is not a cycle and is not a bipartite graph.Remark 5.3.If G is a semiregular connected bipartite graph of order n, then the graph L G is regular and bsr L G n or n − 1.We do not know of any example when bsr L G n − 1.
There exist regular graphs that are cospectral with a line graph but are not line graphs.The complete list of such graphs is given in the following theorem.Proof.The Petersen graph P is the block graph the quasisymmetric 6, 10, 5, 3, 2 -design, so bsr P 6.We also have bsr CP 7 8 Example 4.7 .For n 4, 5, and 6, CP n is an induced subgraph of CP 8 , so bsr CP n ≤ 8.
Let bsr Γ n ≤ 9, and let a 2-distance set X represent Γ in H n .Let h 1 < h 2 be the nonzero distances in X, and let X lie on a sphere of radius k centered at 0. Let f be an isomorphism from Γ to Γ X .For each vertex x of Γ we regard f x as a k-subset of n .
Suppose Γ is not a line graph.Since the least eigenvalue of Γ is −2, Theorem 3.4 implies that h 2 2h 1 .Since n ≤ 9, we assume that k ≤ 4. Proposition 2.13 implies that k / 2, so k 3 or 4. By Theorem 5.5, Γ contains one of the nine graphs of Figure 1 as an induced subgraph.All subgraphs of Γ considered throughout the proof are assumed to be induced subgraphs.Suppose Γ contains F 3 as a subgraph.Suppose also that every subgraph of order 3 of Γ has at least two edges.We assume without loss of generality that f q 1234 and f s 5678.Let x be a vertex of Γ, x / q and x / s.Since the subgraph with the vertex set qsx has at least two edges, x is adjacent to both q and s.Therefore, f x is a 4-subset of 12345678 for every vertex x of Γ.This implies that x is not adjacent to at most one other vertex.Since Γ is regular and not complete, Γ is a cocktail party graph CP m .Therefore, m 1 ≤ bsr Γ ≤ 9 and bsr Γ / m 1 for m 8 Example 4.7 .Since CP 2 and CP 3 are line graphs of C 4 and K 4 , resp., we have 4 ≤ n ≤ 7.

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Case 2 h 1 2 and k 3 .Suppose Γ contains F 1 as a subgraph.We let f p 123, f q 124, f r 135, and f s 236.Since the degree of q in F 1 is 1 and the degree of p is 3, Γ has vertices q 1 and q 2 adjacent to q but not to p. Then f q 1 146 and f q 2 245.Similarly, we find vertices r 1 and r 2 adjacent to r but not to p and vertices s 1 and s 2 adjacent to s but not to p and assume that f r 1 345, f r 2 156, f s 1 256, and f s 2 346.The set U of the 10 vertices that we have found is the vertex set of a Petersen subgraph of Γ.The set f U consists of ten 3-subsets of 123456, no two of which are disjoint.Therefore, if Γ has a vertex v / ∈ U, then f v is disjoint from at least one of the sets f x , x ∈ U; a contradiction.Thus, Γ is the Petersen graph.
If Γ contains F 2 as a subgraph, we let f p 123 and f r 145.Then f q , f s , f t ∈ {124, 125, 134, 135}.Since |f q ∩ f s | |f t ∩ f s | 1, there is no feasible choice for f s .
If Γ contains F 3 as a subgraph, we assume that f p 123, f q 124, and f s 135.Then f r , f t ∈ {125, 134}, and therefore |f r ∩ f t | / 2.
Let Γ contains F 4 as a subgraph.Suppose first that f p ∩ f q ∩ f s / ∅.Then we assume that f p 123, f q 145, f s 167, f t 124, and f u 136, and there is no feasible choice for f r .Suppose now that f p ∩ f q ∩ f s f r ∩ f q ∩ f s ∅.We assume that f p 123, f q 145, and f s 246.Then f t 125 or 134.If f t 125, then f u 234 and f r 235.Γ has distinct vertices q 1 and q 2 adjacent to q but not to r, and we have f q 1 f q 2 156; a contradiction.If f t 134, then f u 126 and f r 136.Γ has distinct vertices q 1 and q 2 adjacent to q but not to t, and we have again f q 1 f q 2 156.If Γ contains F 5 as a subgraph, we let f p 123, f q 145, and f t 124.Then f r 234 or 126, and in either case f s f u .If Γ contains F 6 as a subgraph, we let f p 123, f q 124, and f s 135.Then we assume that f r 125.This implies f u 235 and f t 126.Γ has distinct vertices q 1 and q 2 adjacent to q but not to r.Then f q 1 134 and f q 2 234, so both q 1 and q 2 are adjacent to p. Since q 1/ ∼u and q 2 ∼ u, Γ has three distinct vertices u i adjacent to u but not to p.However, f u i 245 for all these vertices.If Γ contains F 8 as a subgraph, we let f p 123 and f q 124 and assume that f s 156 or 345.If f s 156, then f r 135, f u 136, and there is no feasible choice for f t .If f s 345, then f r 135, f u 235, and again there is no feasible choice for f t .Suppose Γ contains F 7 or F 9 as a subgraph.We let f p 123, f q 124, and f t 135.Then f r 146 or 245 and f s 156 or 345, respectively.In either case, there is no feasible choice for f u .
Case 3 h 1 2 and k 4 .Suppose Γ contains F 1 as a subgraph.We let f p 1234, f q 1235, f r 1246, and f s 1347.Γ has vertices q 1 and q 2 adjacent to q but not to p. Then f q 1 1257 and f q 2 1356.Similarly, we find vertices r 1 and r 2 adjacent to r but not to p and vertices s 1 and s 2 adjacent to s but not to p and assume that f r 1 1267, f r 2 1456, f s 1 1367, and f s 2 1457.The ten vertices that we have found form a Petersen subgraph of Γ.If 1 ∈ f x for every vertex x of Γ, then we delete 1 from each f x and refer to Case 2. Suppose that there is a vertex x with 1 / ∈ f x .Then the 4-set f x must meet each of the sets 234, 235, 246, 347, 257, 356, 267, 456, 367, and 457 in at least two points.Thus, there is no feasible choice for f x .

The distance h a, b between a and b b 1
, b 2 , . . ., b n ∈ H n is the number of indices i for which a i / b i .The Euclidean norm of a vector x ∈ R n is denoted as x , so, for a, b ∈ H n , h a, b a − b 2 .A set X ⊂ H n is called a 2-distance set if |{h a, b : a, b ∈ X, a / b}| 2.A sphere with center c ∈ H n and integer radius k, 1 ≤ k ≤ n − 1, is the set of all points x ∈ H n such that h c, x k.Any subset of a sphere of radius k is called a spherical set of radius k .

Theorem 3 . 3 .
Let Γ be a connected regular graph, and let m be the multiplicity of the least eigenvalue of Γ.Then bsr Γ ≥ |Γ| − m.Moreover, bsr Γ |Γ| − m if and only if Γ is the block graph of a nonsquare nonsingular quasisymmetric 1-design.
fact, 3.15 must hold whenever bsr Γ < |Γ|, because otherwise rank A ≥ |Γ| − 1 and then bsr Γ ≥ |Γ|.If bsr Γ |Γ| and 3.15 does not hold, then σ 0 0. It has also been shown that if bsr Γ |Γ| − m, then the replication number of the corresponding 1-design is |Γ|k/ b − m .We combine these observations in the following two theorems.Theorem 3.4.Let Γ be a connected regular graph of order b and degree d, and let m be the multiplicity of the least eigenvalue ρ of Γ.Let bsr Γ b − m, and let Γ be isomorphic to Γ X , where X is a 2-subset of H b−m lying on a sphere of radius k centered at 0

Example 3 . 7 .
Let D be a v, b, r, k, 1 -design with b ≥ v r and k ≥ 3, and let D be an incidence structure obtained by deleting from D one point and all blocks containing this point.Then D is a 1-design with v − 1 points, b − r > v − 1 blocks of cardinality k, replication number r − 1, and intersection numbers 0 and 1.Without loss of generality, we assume that the point set of D is v , the deleted point is v, and the deleted blocks are{1, 2, . . ., k − 1, v}, {k, k 1, . . ., 2k − 2, v}, . . ., {v − k 1, v − k 2, . . ., v}.3.16Let N be the corresponding incidence matrix of D .Then NN is an r × r block matrix of k − 1 × k − 1 blocks with all diagonal blocks equal to r − 1 I and all off-diagonal blocks equal J.The spectrum of NN consists of eigenvalues r − 1 k of multiplicity 1, r − 1 of multiplicity k − 2 r, and r − k of multiplicity r − 1.Therefore, det NN / 0; that is, the design D is nonsingular.The spectrum of N N is obtained by adjoining the eigenvalue 0 of multiplicity b−r −v 1 to the spectrum of NN .Since N N kI A, where A is an adjacency matrix of the block graph Γ of D , we determine that the multiplicities of the largest and the smallest eigenvalues of A are 1 and b − r − v 1, respectively.Therefore, Γ is a connected regular graph and bsr Γ v − 1.

Theorem 4. 6 .
Let Γ be a connected strongly regular graph of order b, and let m be the multiplicity of the least eigenvalue of Γ.Then bsr Γ b − m if and only if Γ is the block graph of a quasisymmetric 2-design.Proof.If Γ is the block graph of a quasisymmetric 2-design D, then Remark 2.6 and Theorem 3.3 imply that bsr Γ b − m.Suppose now that bsr Γ b − m, and let X be a spherical 2-distance subset of H b−m representing Γ.Let h 1 < h 2 be the nonzero distances in X and k the radius of the sphere centered at 0 and containing X. Every block of the incidence structure D X b − m , B is of cardinality k, the intersection numbers of D are α k − h 2 /2 < β k − h 1 /2, and the replication number of D is r bk/ b − m Theorem 3.4 .The graph Γ is the block graph of D

Theorem 5 . 1 .
If Γ is the line graph of a connected regular graph of order n ≥ 4, then bsr Γ n.

Theorem 5 .4 see 11 .Theorem 5 . 6 .
Let a regular graph Γ be cospectral with the line graph L G of a connected graph G.If Γ is not a line graph, then G is a regular 3-connected graph of order 8 or K 3,6 or the semiregular bipartite graph of order 9 with 12 edges.Since bsr L G < 10 for every graph G listed in Theorem 5.4, the next theorem implies that if a connected regular graph Γ is cospectral with a line graph L G and if bsr Γ bsr L G , then Γ is a line graph.The proof is based on the following theorem according to Beineke 12 .Theorem 5.5.A graph is a line graph if and only if it does not contain as an induced subgraph any of the nine graphs ofFigure 1.Let the least eigenvalue of a connected regular graph Γ be equal to −2.If bsr Γ < 10, then Γ is a line graph or the Petersen graph or CP n with 4 ≤ n ≤ 7.

Figure 1 Case 1 h
Figure 1

Proposition 2.11. Every
Definition 2.10.One will say that a spherical 2-distance set X ⊂ H n represents a graph Γ in H n if Γ is isomorphic to Γ X .The least n for which such a set X exists is called the binary spherical representation number of Γ and is denoted as bsr Γ .simple graph Γ, except null graphs and complete graphs, admits a spherical representation in H n if n is sufficiently large.Let Γ be a noncomplete graph of order b with e ≥ 1 edges, and let N n ij be an incidence matrix of Γ.For i 1, 2, . . ., b, let X i {j ∈ e : n ij 1}.Let k max{|X i | : 1 ≤ i ≤ b}, and let Y 1 , Y 2 , . . ., Y b be pairwise disjoint subsets of {e 1, e 2, . . ., e bk} such that |Y i | k − |X i |.For i 1, 2, . . ., b, let x i then let Y be the set of rows of the matrix J − A. The distance between two distinct points of Y is equal to 2 d − μ if the points correspond to adjacent vertices of Γ; otherwise, it is equal to 2 d − λ .Therefore, Y is a 2-distance set in H n , lying on a sphere of radius n − d centered at O, and Γ is isomorphic to Γ Y .This proposition and Corollary 3.6 imply the next result.If the least eigenvalue of a strongly regular graph Γ of order n is irrational, then bsr Γ n.The least eigenvalue of a strongly regular graph is irrational if and only if it is an srg n, n − 1 /2, n − 5 /4, n − 1 /4 , where n ≡ 1 mod 4 is not a square.A graph with these parameters exists if and only if there exists a conference matrix of order n 1.