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International Journal of Combinatorics

Volume 2011 (2011), Article ID 101928, 15 pages

http://dx.doi.org/10.1155/2011/101928

## Binary Representations of Regular Graphs

Department of Mathematics, Central Michigan University, Mount Pleasant, MI 48859, USA

Received 17 January 2011; Accepted 2 August 2011

Academic Editor: Hajo Broersma

Copyright © 2011 Yury J. Ionin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

For any 2-distance set in the *n*-dimensional binary Hamming space , let be the graph with 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 .
The binary spherical representation number of a graph , or bsr(), is the least *n* such that is isomorphic to , where is a 2-distance set lying on a sphere in . It is shown that if is a connected regular graph, then bsr, where *b* is the order of and *m* is the multiplicity of the least eigenvalue of , and the case of
equality is characterized. In particular, if is a connected strongly regular graph, then
bsr if and only if 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.

#### 1. 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 ** is the multiplicity of the least eigenvalue of a connected strongly regular graph ** of order **, then the vertex set of ** can be represented as a set of points, lying on a sphere in **, so that there exist positive real numbers ** such that the distance between any two distinct vertices is equal to ** if they are adjacent as vertices of ** and it is equal to ** otherwise.* This result was recently generalized to all connected regular graphs in [2]. It has also been proved in [2] that, given and , such a representation of a connected regular graph in 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 graph can be represented, is at least , where and have the same meaning as in the previous paragraph.

It is also well known that the block graph of a quasisymmetric 2-design is strongly regular. However, many strongly regular graphs are not block graphs, and there is no good characterization of the graphs that are block graphs of quasisymmetric 2-designs. The situation changes if we consider the representation of graphs in binary Hamming spaces. We will show (Theorem 4.6) that a connected strongly regular graph admits a representation in the binary Hamming space of the minimal dimension if and only if it is the block graph of a quasisymmetric 2-design.

At the dawn of graph theory there was a short-lived conjecture that a graph is determined by the spectrum of its adjacency matrix. Of course, it is not true (see a very interesting discussion in [3]). However, some classes of graphs can be described by their spectra. In particular, if a connected regular graph has the same spectrum as a line graph, then it is almost always a line graph itself (all exceptions are known). We will show (Corollary 5.7) that if a connected regular graph is cospectral with a line graph of a graph and, beside that, the minimal dimension of a binary Hamming space, in which either graph can be represented, is the same for and , then is a line graph.

#### 2. Preliminaries

All graphs in this paper are finite and simple, and all incidence structures are without repeated blocks. For a graph , denotes the * order of *, that is, the number of vertices. If and are vertices of a graph , then means that and are adjacent, while means that and 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 to denote identity matrices and to denote square matrices with every entry equal to 1. The order of and will be always apparent from the context. We denote as and 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 instead of .

If is a positive integer, then denotes the set .

*Definition 2.1. *The *binary Hamming space * consists of all -tuples with each equal to 0 or 1. When it is convenient, one identifies with the set . The distance between and is the number of indices for which . The Euclidean norm of a vector is denoted as , so, for , .

A set is called a * 2-distance set* if .

A * sphere* with center and integer radius , , is the set of all points such that . Any subset of a sphere (of radius ) is called a * spherical set* (*of radius *).

*Remark 2.2. *The sphere of radius in , centered at , coincides (as a set) with the sphere of radius centered at the opposite point . This allows us to assume, when needed, that the radius of a sphere does not exceed . A sphere of radius in centered at , regarded as a subset of , is the intersection of the unit cube and the hyperplane .

*Remark 2.3. *For , the distance between any two points of a spherical set in is even.

*Definition 2.4. *An *incidence structure* (without repeated blocks) is a pair , where is a nonempty finite set (of *points*) and is a nonempty set of subsets of (*blocks*). The cardinality of the intersection of two distinct blocks is called an *intersection number* of . An incidence structure is said to be *quasisymmetric* if it has exactly two distinct intersection numbers. For a nonnegative integer , an incidence structure is called a -*design* if all blocks of have the same cardinality and every set of points is contained in the same number of blocks. A -design with an (points versus blocks) incidence matrix is called *nonsquare *if is not a square matrix, and it is called *nonsingular* if . A 2-design is also called a -design, where is the number of points, is the number of blocks, is the *replication number*, that is, the number of blocks containing any given point, is the block size, and is the number of blocks containing any given pair of points.

With any quasisymmetric incidence structure we associate its * block graph*.

*Definition 2.5. *If is a quasisymmetric incidence structure with intersection numbers , then the *block graph* of is the graph whose vertices are the blocks of and two vertices are adjacent if and only if the corresponding blocks meet in points.

*Remark 2.6. *If a regular graph, other than a complete graph, is connected, then it has at least three distinct eigenvalues. It is strongly regular if and only if it has exactly three distinct eigenvalues. If is a quasisymmetric 2-design, then it is nonsquare and its block graph is strongly regular. If is a quasisymmetric -design with block size and intersection numbers , then , where is an incidence matrix of and is an adjacency matrix of the block graph of . If is a -design, then . Therefore, det, so is nonsingular. For these and other basic results on designs and regular graphs, see [1] or [4].

*Definition 2.7. *Let be a 2-distance set of cardinality in , and let be the nonzero distances in . One denotes as the graph whose vertex set is and the edge set is the set of all pairs with . For , let and , so is the characteristic vector of . Let . One denotes as the incidence structure .

*Remark 2.8. *If is a spherical 2-distance set centered at , then the incidence structure is a quasisymmetric 0-design and is its block graph.

Proposition 2.9. *Let be a 2-distance set in , and let be the nonzero distances in . If the graph is connected, then .*

*Proof. *Suppose . If , and are distinct vertices of such that and , then the triangle inequality implies that . Therefore, all neighbors of form a connected component of . Since is not a complete graph, it is not connected; a contradiction.

*Definition 2.10. *One will say that a spherical 2-distance set *represents a graph * *in * if is isomorphic to . The least for which such a set exists is called the *binary spherical representation number of * and is denoted as .

Proposition 2.11. *Every simple graph , except null graphs and complete graphs, admits a spherical representation in if is sufficiently large.*

*Proof. *Let be a noncomplete graph of order with edges, and let be an incidence matrix of . For , let . Let , and let be pairwise disjoint subsets of such that . For , let , where if and only if . Then, for , the distance between points and is equal to if the th and th vertices of are adjacent, and it is equal to otherwise. Since is not a complete graph, the set is a 2-distance set representing in , and this set lies on a sphere of radius centered at .

If the graph in the above proof is regular, we do not need to add columns to its incidence matrix .

Proposition 2.12. *If is a noncomplete regular graph with edges, then .*

Theorem 5.1 implies that if is a cycle, then its binary spherical representation number equals the number of edges.

For any graph , the * line graph of *, denoted as , is the graph whose vertex set is the edge set of ; two distinct vertices of are adjacent if and only if the corresponding edges of share a vertex. Line graphs are precisely the graphs representable by spherical 2-distance sets of radius 2.

Proposition 2.13. *A graph can be represented in by a spherical 2-distance sets of radius 2 if and only if is isomorphic to the line graph of a graph of order .*

*Proof. *If , where is a graph of order , then the columns of an incidence matrix of form a 2-distance subset of of radius 2 representing . Conversely, let be a 2-distance subset of of radius 2 centered at and representing a graph . Let be a graph whose incidence matrix coincides with an incidence matrix of . Then and is isomorphic to .

*Remark 2.14. *Let be a regular graph of degree , and let be the set of columns of an incidence matrix of . Then is a quasisymmetric 1-design (with block size 2 and replication number ) and is its incidence matrix. If , 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 is an incidence matrix of a graph of order and is the number of connected components of , then
*

#### 3. Lower Bounds

The main tool in obtaining a lower bound on is the following classical theorem of distance geometry.

*Definition 3.1. *Let be a set of points in *. *The* Schoenberg matrix of ** with respect to a point * is the matrix of order with

Theorem 3.2 (see [6, 7]). *If is a finite set in , then, for any , the Schoenberg matrix is positive semidefinite and .*

We will now derive a sharp lower bound on the binary spherical representation number of a connected regular graph.

Theorem 3.3. *Let be a connected regular graph, and let be the multiplicity of the least eigenvalue of . Then . Moreover, if and only if is the block graph of a nonsquare nonsingular quasisymmetric 1-design.*

*Proof. *Let , and let be isomorphic to , where is a spherical 2-subset of . Let be the nonzero distances in and the radius of a sphere in containing . Without loss of generality, we assume that this sphere is centered at . Then is the center of an Euclidean sphere containing . The radius of this sphere is equal to . Let be an adjacency matrix of . Then the matrix
is the Schoenberg matrix of the set with respect to .

Let be the degree of , all distinct eigenvalues of , and their respective multiplicities. Then the eigenvalues of are
with as an eigenvector, and, for ,
(with eigenvectors orthogonal to ). For , the multiplicity of is at least . (It is greater than if , .)

Theorem 3.2 implies that all eigenvalues of are nonnegative, so for . Therefore, . On the other hand, since both and lie in the hyperplane , Theorem 3.2 implies that , so .

Suppose now that . Then , and therefore . From we derive

The incidence structure has points, blocks, all of cardinality , and two intersection numbers, . The graph is the block graph of . Using and , we transform (3.5) into

For each , let denote the number of blocks of containing . Fix a block and count in two ways pairs , where , , and :
Using this equation and (3.6), we derive
Therefore,
Since each contributes into the left-hand side of this equation, we obtain that

On the other hand, counting in two ways pairs with and yields
Thus,
Therefore, for . Thus, is a quasisymmetric 1-design. (Note that we have derived this result from (3.5) rather than from a stronger equation .) Since , the 1-design is non-square, so we have to show that it is nonsingular.The incidence matrix of satisfies the equation
Therefore, the eigenvalues of are
Since and since , we obtain that and for . Since the multiplicity of is the same as the multiplicity of , we have . Therefore, , and then , that is, is nonsingular.

Suppose now that is the block graph of a nonsquare nonsingular quasisymmetric 1-design with intersection numbers . The design has less points than blocks, so let be the number of blocks and the number of points. We have to show that is the multiplicity of the least eigenvalue of and that .

Let be an incidence matrix of and the set of all columns of regarded as points in . Then is a 2-distance set and is . The set lies on a sphere of radius centered at , where is the cardinality of each block of , and the nonzero distances in are and .

Matrix satisfies (3.13) with being an adjacency matrix of . Let be all distinct eigenvalues of . Then the eigenvalues of are given by (3.14). Since has more rows than columns, we have . Since , the sum of the multiplicities of the nonzero eigenvalues of is , so the multiplicity of is equal to . Therefore, the multiplicity of is equal to , and then . Since is in , we have .

It has been shown in the course of this proof that if , then , which implies (3.5), and , which implies In fact, (3.15) must hold whenever , because otherwise and then . If and (3.15) does not hold, then . It has also been shown that if , then the replication number of the corresponding 1-design is . We combine these observations in the following two theorems.

Theorem 3.4. *Let be a connected regular graph of order and degree , and let be the multiplicity of the least eigenvalue of . Let , and let be isomorphic to , where is a 2-subset of lying on a sphere of radius centered at . Let be the nonzero distances in . Then,*(i)*;*(ii)*;*(iii)* is a nonsquare nonsingular quasisymmetric 1-design with points, blocks, block size , replication number , and intersection numbers and .*

Theorem 3.5. *Let be a connected regular graph of order and degree , and let be the least eigenvalue of . Let , and let be isomorphic to , where is a 2-subset of lying on a sphere of radius . Let be the nonzero distances in :*(i)*if , then is a quasisymmetric 1-design;*(ii)*if , then ;*(iii)*if and , then .*

If , then is rational, so (ii) implies that the following useful result.

Corollary 3.6. *If the least eigenvalue of a connected regular graph is irrational, then .*

An infinite family of regular graphs attaining the lower bound of Theorem 3.3 is given in the following example.

*Example 3.7. *Let be a -design with and , and let be an incidence structure obtained by deleting from one point and all blocks containing this point. Then is a 1-design with points, blocks of cardinality , replication number , and intersection numbers 0 and 1. Without loss of generality, we assume that the point set of is , the deleted point is , and the deleted blocks are
Let be the corresponding incidence matrix of . Then is an block matrix of blocks with all diagonal blocks equal to and all off-diagonal blocks equal . The spectrum of consists of eigenvalues of multiplicity 1, of multiplicity , and of multiplicity . Therefore, ; that is, the design is nonsingular. The spectrum of is obtained by adjoining the eigenvalue 0 of multiplicity to the spectrum of . Since , where is an adjacency matrix of the block graph of , we determine that the multiplicities of the largest and the smallest eigenvalues of are 1 and , respectively. Therefore, is a connected regular graph and .

#### 4. Strongly Regular Graphs

For strongly regular graphs we first obtain a sharp upper bound for the binary spherical representation number.

Proposition 4.1. *If is a connected strongly regular graph of order , then .*

*Proof. *Let be an , and let be an adjacency matrix of . Then . Therefore, . Let be the set of rows of regarded as points in . Then the distance between two distinct points of is equal to if the points correspond to adjacent vertices of ; otherwise, it is equal to . Thus, is a 2-distance set in , lying on a sphere of radius centered at , and, if , then is isomorphic to .

If , then let be the set of rows of the matrix . The distance between two distinct points of is equal to if the points correspond to adjacent vertices of ; otherwise, it is equal to . Therefore, is a 2-distance set in , lying on a sphere of radius centered at , and is isomorphic to .

This proposition and Corollary 3.6 imply the next result.

Corollary 4.2. *If the least eigenvalue of a strongly regular graph of order is irrational, then .*

*Remark 4.3. *The least eigenvalue of a strongly regular graph is irrational if and only if it is an , where is not a square. A graph with these parameters exists if and only if there exists a conference matrix of order .

*Example 4.4. *Let be the complement of the cycle . The least eigenvalue of is irrational, so . Suppose , and let be isomorphic to , where is a 2-subset of with nonzero distances , lying on a sphere of radius centered at . Since and are even, and (Proposition 2.9), we have , or , . In either case, Theorem 3.5(iii) yields an equation without integer solutions. Thus, , so the strong regularity in Proposition 4.1 is essential.

*Remark 4.5. *There are 167 nonisomorphic strongly regular graphs with parameters [8]. The least eigenvalue of these graphs is −6 of multiplicity 18. Theorem 3.3 and Proposition 4.1 imply that if is any of these 167 graphs, then . 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 [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.

Theorem 4.6. *Let be a connected strongly regular graph of order , and let be the multiplicity of the least eigenvalue of . Then if and only if is the block graph of a quasisymmetric 2-design.*

*Proof. *If is the block graph of a quasisymmetric 2-design , then Remark 2.6 and Theorem 3.3 imply that .

Suppose now that , and let be a spherical 2-distance subset of representing . Let be the nonzero distances in and the radius of the sphere centered at and containing . Every block of the incidence structure is of cardinality , the intersection numbers of are , and the replication number of is (Theorem 3.4). The graph is the block graph of . Let be the eigenvalues of . Since is connected, the multiplicity of is 1. Since the multiplicity of is , the multiplicity of is .

Let be an adjacency matrix of . Theorem 3.4(ii) implies that . Since , we use Theorem 3.4(i) to derive that
where . Since , we have .

Let be an incidence matrix of . Then

From these equations we determine the eigenvalues of : , , and . Their respective multiplicities are 1, , and . Therefore, the eigenvalues of are of multiplicity 1 and of multiplicity . Since , the eigenspace of corresponding to the eigenvalue is generated by . Therefore, is the eigenspace corresponding to the eigenvalue . On the other hand, the matrix has the same eigenvalues with the same respective eigenspaces. Thus, , and therefore is a quasisymmetric 2-design with intersection numbers and . The graph is the block graph of this design.

*Example 4.7. *The *Cocktail Party graph * has vertices split into 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 points (cf. [4, Theorem 8.2.23]). The least eigenvalue of is −2 of multiplicity . By Theorem 4.6, and if and only there exists a Hadamard matrix of order . 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.

#### 5. Line Graphs

In this section we determine the binary spherical representation number for the line graphs of regular graphs. If is an incidence matrix of a graph , then , where is an adjacency matrix of the line graph . Let be connected and have vertices and edges. If , then the least eigenvalue of is 0, and therefore the least eigenvalue of is . Since matrices and have the same positive eigenvalues, Proposition 2.15 implies that the multiplicity of is equal to if the graph is not bipartite, and it is equal to if is a connected bipartite graph. If , then is a cycle, so is a cycle of order too. If is even, then the least eigenvalue of is −2 of multiplicity 1; if is odd, then the least eigenvalue of is irrational. See [9] for details.

Theorem 5.1. *If is the line graph of a connected regular graph of order , then .*

*Proof. *Let be the line graph of a connected regular graph of order and degree . Then is a connected regular graph of order and degree . The columns of an incidence matrix of form a spherical 2-distance set in representing , so .

Suppose first that , that is, is , and that is odd. Then the least eigenvalue of is irrational. Therefore, by Corollary 3.6, so .

Suppose now that and the graph is not bipartite. From Proposition 2.15, the multiplicity of the least eigenvalue of is , and then Theorem 3.3 implies that , so .

Suppose finally that is a bipartite graph (this includes the case with even ). Then the least eigenvalue of is −2 and its multiplicity is . Therefore, by Theorem 3.3, . Suppose . Theorem 3.4(ii) implies that and then the condition (i) of Theorem 3.4 can be rewritten as
If is odd, then and are relatively prime; if is even, then and are relatively prime. In either case, divides . However, . Therefore, .

The graph is the line graph of the bipartite graph with the bipartition sets of cardinality . 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 with more than three vertices is the block graph of a nonsquare nonsingular quasisymmetric 1-design if and only if is not a cycle and is not a bipartite graph.*

*Remark 5.3. *If is a semiregular connected bipartite graph of order , then the graph is regular and or . We do not know of any example when .

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.

Theorem 5.4 (see [11]). *Let a regular graph be cospectral with the line graph of a connected graph . If is not a line graph, then is a regular 3-connected graph of order 8 or or the semiregular bipartite graph of order 9 with 12 edges.*

Since for every graph listed in Theorem 5.4, the next theorem implies that if a connected regular graph is cospectral with a line graph and if , then 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 of Figure 1.*

Theorem 5.6. *Let the least eigenvalue of a connected regular graph be equal to −2. If , then is a line graph or the Petersen graph or with . *

*Proof. *The Petersen graph is the block graph the quasisymmetric -design, so . We also have (Example 4.7). For , and 6, is an induced subgraph of , so .

Let , and let a 2-distance set represent in . Let be the nonzero distances in , and let lie on a sphere of radius centered at . Let be an isomorphism from to . For each vertex of we regard as a -subset of .

Suppose is not a line graph. Since the least eigenvalue of is −2, Theorem 3.4 implies that . Since , we assume that . Proposition 2.13 implies that , so or 4. By Theorem 5.5, contains one of the nine graphs of Figure 1 as an induced subgraph. All subgraphs of considered throughout the proof are assumed to be induced subgraphs.*Case 1 : (. Then , and therefore ). *If contains a coclique of size 3, then . This rules out subgraphs and . If the subgraph induced by on a set of three vertices has only one edge, then . This rules out subgraphs , , , , , and .

Suppose contains as a subgraph. Suppose also that every subgraph of order 3 of has at least two edges. We assume without loss of generality that and . Let be a vertex of , and . Since the subgraph with the vertex set has at least two edges, is adjacent to both and . Therefore, is a 4-subset of 12345678 for every vertex of . This implies that is not adjacent to at most one other vertex. Since is regular and not complete, is a cocktail party graph . Therefore, and for (Example 4.7). Since and are line graphs (of and , resp.), we have .*Case 2 ( and ). *Suppose contains as a subgraph. We let , , , and . Since the degree of in is 1 and the degree of is 3, has vertices and adjacent to but not to . Then and . Similarly, we find vertices and adjacent to but not to and vertices and adjacent to but not to and assume that , , , and . The set of the 10 vertices that we have found is the vertex set of a Petersen subgraph of . The set consists of ten 3-subsets of 123456, no two of which are disjoint. Therefore, if has a vertex , then is disjoint from at least one of the sets , ; a contradiction. Thus, is the Petersen graph.

If contains as a subgraph, we let and . Then . Since , there is no feasible choice for .

If contains as a subgraph, we assume that , , and . Then , and therefore .

Let contains as a subgraph. Suppose first that . Then we assume that , , , , and , and there is no feasible choice for . Suppose now that . We assume that , , and . Then or 134. If , then and . has distinct vertices and adjacent to but not to , and we have ; a contradiction. If , then and . has distinct vertices and adjacent to but not to , and we have again .

If contains as a subgraph, we let , , and . Then or 126, and in either case .

If contains as a subgraph, we let , , and . Then we assume that . This implies and . has distinct vertices and adjacent to but not to . Then and , so both and are adjacent to . Since and , has three distinct vertices adjacent to but not to . However, for all these vertices.

If contains as a subgraph, we let and and assume that or 345. If , then , , and there is no feasible choice for . If , then , , and again there is no feasible choice for .

Suppose contains or as a subgraph. We let , , and . Then or 245 and or 345, respectively. In either case, there is no feasible choice for .*Case 3 ( and ). *Suppose contains as a subgraph. We let , , , and . has vertices and adjacent to but not to . Then and . Similarly, we find vertices and adjacent to but not to and vertices and adjacent to but not to and assume that , , , and . The ten vertices that we have found form a Petersen subgraph of . If for every vertex of , then we delete 1 from each and refer to Case 2. Suppose that there is a vertex with . Then the 4-set must meet each of the sets , and 457 in at least two points. Thus, there is no feasible choice for .

If contains as a subgraph, we let , , and . Then and there is no feasible choice for .

If contains as a subgraph, we assume that , , and . Then , and therefore ; a contradiction.

If contains as a subgraph, we let , , , , and . Then . Let and be vertices of adjacent to but not to . Then and , and there is no feasible choice for , where and .

If contains as a subgraph, we let , , and . Then we may assume that either and or and . In either case, there is no feasible choice for .

Suppose contains as a subgraph. We let and . Then . Since the subgraph induced on is triangle-free, we let , , , and . Let and be distinct vertices of adjacent to but not to . Then , so both and are adjacent to but not to . Therefore, has at least four vertices adjacent to but not to . However, ; a contradiction.

If contains as a subgraph, we let , , and . Then , and there is no feasible choice for .

Suppose contains or as a subgraph. We let , , and . Then and , so we assume that . In each case, there is no feasible choice for .

Corollary 5.7. *Let be a connected regular graph cospectral with a line graph of a connected graph . If , then is a line graph.*

*Proof. *If is not an exceptional graph from Theorem 5.4, then is a line graph by that theorem. If is one of the exceptional graphs, then and has more edges than vertices. Therefore, the least eigenvalue of is −2. Since the Petersen graph and graphs are not exceptional, Theorem 5.6 implies that is a line graph.

*Example 5.8. *Let be the set of all points of with even sum of coordinates. It is a 2-distance set and is the complement of the *Clebsch graph*. The least eigenvalue of is −2, and, since it is not a line graph, Theorem 5.6 implies that (so is not spherical). Let be the set of all points such that is even and for . Then is a spherical 2-distance set and is isomorphic to . Thus, .

*Example 5.9. *The *Shrikhande graph* is cospectral with , and the three *Chang graphs* are cospectral with *,* the line graph of , so we have examples of cospectral strongly regular graphs with distinct binary spherical representation numbers. It can be shown that the binary spherical representation number of the Shrikhande graph is 12.

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