Abstract. Let G be a simple graph of order n. Let c = a + b √

THE CONJUGATE MAIN EIGENVALUES OF VERTEX DELETED SUBGRAPHS OF A STRONGLY REGULAR GRAPH

Mirko Lepovi´ c

(Received February 25, 2010)

Abstract. Let G be a simple graph of order n. Let c = a + b √

m and c = a − b √

m, where a and b are two nonzero integers and m is a positive integer such that m is not a perfect square. We say that A

= [c

] is the conjugate adjacency matrix of the graph G if c

= c for any two adjacent vertices i and j, c

= c for any two nonadjacent vertices i and j, and c

= 0 if i = j.

Let (A

)

= [c

] for any nonnegative integer k. Further, let G be a strongly regular graph of degree r ≥ 1, understanding that G is not the complete graph.

Then c

= τ

for any two adjacent vertices i and j and c

= θ

for any two distinct nonadjacent vertices i and j, where τ

and θ

are two fixed real numbers. Let r

= rc + ((n − 1) − r) c. We demonstrate that

µ

= τ

− θ

+ c − c
r

± q

τ

− θ

− (c − c) r

+ 4(c − c) c τ

− c θ

2 c − c
,

where µ

and µ

are the conjugate main eigenvalues of its vertex deleted subgraphs G

= G r i for i = 1, 2, . . ., n.

Let G be a simple graph of order n. The spectrum of G consists of the eigenvalues λ ₁ ≥ λ ₂ ≥ · · · ≥ λ _n of its (0,1) adjacency matrix A = A(G) and is denoted by σ(G).

Let c = a + b √

m and c = a − b √

m where a and b are two nonzero integers and m is a positive integer such that m is not a perfect square. We say that A ^c = [c ij ] is the conjugate adjacency matrix of G if c ij = c for any two adjacent vertices i and j, c ij = c for any two nonadjacent vertices i and j, and c ij = 0 if i = j. The conjugate spectrum of G consists of the eigenvalues λ ^c ₁ ≥ λ ^c ₂ ≥ · · · ≥ λ ^c _n of its conjugate adjacency matrix A ^c = A ^c (G) and is denoted by σ ^c (G).

Let i be a fixed vertex from the vertex set V (G) = {1, 2, . . ., n} and let G i = Gri be its corresponding vertex deleted subgraph. Let S i denote the neighborhood of i, defined as the set of all vertices of G which are adjacent to i. Let d i denote the degree of the vertex i and let ∆ i = P

j∈S

d j . Besides, let A ^k = [a ^(k) _ij ] for any nonnegative integer k.

Proposition 1 (Lepovi´ c [4]). Let G be a connected or disconnected graph of order n. Then for any vertex deleted subgraph G i we have:

(1 ⁰ ) ∆ j (G i ) = ∆ j (G) − d i (G) − a ⁽²⁾ _ij (G) if j ∈ S i ; (2 ⁰ ) ∆ j (G i ) = ∆ j (G) − a ⁽²⁾ _ij (G)

if j ∈ T i = V (G i ) r S ⁱ .

2010 Mathematics Subject Classification 05C50.

Key words and phrases: strongly regular graph, conjugate adjacency matrix, conjugate main

eigenvalue.

We say that a regular graph G of order n and degree r ≥ 1 is strongly regular if there exist non-negative integers τ and θ such that |S i ∩ S j | = τ for any two adjacent vertices i and j, and |S i ∩ S j | = θ for any two distinct non-adjacent vertices i and j, understanding that G is not the complete graph K n . We know that a regular connected graph is strongly regular if and only if it has exactly three distinct eigenvalues.

Theorem 1 (Lepovi´ c [4]). A regular graph G of order n and degree r ≥ 1 is strongly regular if and only if its vertex deleted subgraphs G i have exactly two main ¹ eigenvalues for i = 1, 2, . . ., n.

Theorem 2 (Lepovi´ c [4]). Let G be a connected or disconnected strongly regular graph of order n and degree r. Then for any vertex deleted subgraph G _i we have

µ _1,2 = τ − θ + r ± q

τ − θ − r
²

− 4 θ

2 , (1)

where µ 1 and µ 2 are the main eigenvalues of G i .

Further, let (A ^c ) ^k = [c ^(k) _ij ] for any nonnegative integer k. Let d _i ^c = P n

j=1 c ij and let ∆ ^c _i = P n

j=1 c ⁽²⁾ _ij for i = 1, 2, . . ., n. Of course, d _i ^c = cd i + ((n − 1) − d i )c for i = 1, 2, . . ., n. Since 2b √

m A = − c (J − I) + A ^c , where J is a square matrix of order n with entries equal to 1, by a straight-forward calculation we get

4mb ² a ⁽²⁾ _ij = c ² n − 2 + δ ij
− c d _i ^c + d _j ^c − 2c ij
+ c ⁽²⁾ _ij , (2) where δ ij is the Kronecker delta symbol. Since a ⁽²⁾ _ij = |S i ∩ S j | for i, j = 1, 2, . . ., n, we note that a ⁽²⁾ _ij = a ⁽²⁾ _ik and a ij = a ik if and only if c ⁽²⁾ _ij = c ⁽²⁾ _ik and c ij = c ik if G is regular. Consequently, if G is a strongly regular graph then there exits two real values τ ^c and θ ^c such that τ ^c = c ⁽²⁾ _ij for any two adjacent vertices i and j, and θ ^c = c ⁽²⁾ _ij for any two distinct nonadjacent vertices i and j. In particular, using (2) we find that

τ ^c = 4mb ² τ − r + 1
+ (n − 2) a ² + mb ²
− 2ab n − 2r
√

m , (3)

θ ^c = 4mb ² θ − r
+ (n − 2) a ² + mb ²
− 2ab n − 2 − 2r
√

m . (4)

Let A ^c = A ^c (G) and let (A ^c ) ^k = [c ^(k) _ij ] for any nonnegative integer k, where G denotes the complement of G. Since c ^(k) _ij = c ^(k) _ij for i, j = 1, 2, . . ., n and any defined k, we obtain the following result.

Proposition 2. Let G be a strongly regular graph of order n and degree r with parameters r ^c = cr +((n−1)−r)c and τ ^c and θ ^c defined by (3) and (4), respectively.

Then G is the strongly regular graph with parameters r ^c (G) = r ^c (G), τ ^c (G) = θ ^c (G) and θ ^c (G) = τ ^c (G).

Definition 1. A strongly regular graph of order 4n + 1 and degree r = 2n with τ = n − 1 and θ = n is called the conference graph.

1 We say that µ ∈ σ(G) is the main eigenvalue if and only if hj, Pji = n cos

α > 0, where j is the

main vector (with coordinates equal to 1) and P is the orthogonal projection of the space R

onto

the eigenspace E

(µ). The value β = | cos α| is called the main angle of µ.

Proposition 3. Let G be a strongly regular graph of order n and degree r. If τ ^c (G) = τ ^c (G) then G is a conference graph.

Proof. Using (3) we obtain that τ − r + 1 = τ − r + 1 and n − 2r = n − 2r, where τ = τ (G) and r = (n − 1) − r. In view of this we have τ = τ and r = r. Since τ ^c (G) = θ ^c (G) and τ ^c (G) = τ ^c (G), using (3) and (4) we obtain that τ −r+1 = θ−r.

Finally, since

a ⁽²⁾ _ii + X

j∈S

a ⁽²⁾ _ij + X

j∈T

a ⁽²⁾ _ij = r + rτ + rθ = r ² ,

we find that 2θ = r, which provides the proof. 

Proposition 4. Let G be a strongly regular graph of order n and degree r. If θ ^c (G) = θ ^c (G) then G is a conference graph.

Proof. Let θ ^c (G) = θ ^c (G). Then θ ^c (G) = θ ^c (G) which means that τ ^c (G) = τ ^c (G).

Using Proposition 3 we obtain the statement. 

Proposition 5. Let G be a connected or disconnected graph of order n. Then for any vertex deleted subgraph G i we have:

(1 ⁰ ) d _j ^c (G i ) = d _j ^c (G) − c if j ∈ S i , (2 ⁰ ) d _j ^c (G _i ) = d _j ^c (G) − c if j ∈ T _i .

Proof. Since d j (G i ) = d j (G) − 1 if j ∈ S i and d j (G i ) = d j (G) if j ∈ T i , we obtain the assertion using the fact that d _i ^c = cd i + ((n − 1) − d i )c.  Proposition 6. Let G be a connected or disconnected graph of order n. Then for any vertex deleted subgraph G i we have:

(1 ⁰ ) ∆ ^c _j (G i ) = ∆ ^c _j (G) − c d _i ^c (G) − c ⁽²⁾ _ij (G) if j ∈ S i , (2 ⁰ ) ∆ ^c _j (G _i ) = ∆ ^c _j (G) − c d _i ^c (G) − c ⁽²⁾ _ij (G) if j ∈ T _i . Proof. Since ∆ i = P n

j=1 a ⁽²⁾ _ij for i = 1, 2, . . ., n, using relation (2) by an easy calculation we get ²

∆ ^c _i = 2mb ² 2∆ i − (n − 2)d i − 2e
+ (n − 1) ² (a ² + mb ² )

− 2ab (n − 1) ² − (n − 2)d i − 2e
√m ,

where e is the number of edges of G. Applying the last relation to G _i and making use of Proposition 1, by a straight-forward calculation we obtain the statement.  Theorem 3 (Hagos [2]). A non-regular graph G of order n has exactly two main eigenvalues if and only if there exist two real constants p and q such that ∆ _i + p d _i + q = 0 for i = 1, 2, . . ., n.

Using the same procedure applied in [2] for getting Theorem 3, one can obtain the following result.

2 Of course, if G is a regular graph of order n and degree r we have ∆

= (r

)

. In the case that

G is a regular graph of order 4n + 1 and degree r = 2n we get r

= 4na and ∆

= 16n

a

.

Proposition 7. A non-regular graph G of order n has exactly two conjugate ³ main eigenvalues if and only if there exist two real constants p ^c and q ^c such that ∆ ^c _i + p ^c d _i ^c + q ^c = 0 for i = 1, 2, . . ., n.

Theorem 4. Let G be a connected or disconnected strongly regular graph of order n and degree r. Then for any vertex deleted subgraph G i we have

µ ^c _1,2 =

τ ^c − θ ^c + c − c
r ^c ± q

τ ^c − θ ^c − (c − c) r ^c
2

+ 4(c − c) c τ ^c − c θ ^c

2 c − c
,

where µ ^c ₁ and µ ^c ₂ are the conjugate main eigenvalues of G i .

Proof. Since d _j ^c (G i ) = r ^c − c if j ∈ S i and d _j ^c (G i ) = r ^c − c if j ∈ T i , it is not difficult to see that the following relation

c ⁽²⁾ _ij − c ⁽²⁾ _ik = − τ ^c − θ ^c

c − c d _j ^c (G i ) − d _k ^c (G i )

(5) is satisfied for i = 1, 2, . . ., n, keeping in mind that c ⁽²⁾ _ij = τ ^c if j ∈ S _i and c ⁽²⁾ _ij = θ ^c if j ∈ T i . Let us assume ⁴ that c ⁽²⁾ _ik = θ ^c . Using (5), we get

c ⁽²⁾ _ij = − τ ^c − θ ^c

c − c d _j ^c (G i ) + (τ ^c − θ ^c ) r ^c − (c τ ^c − c θ ^c )

c − c . (6)

Case 1. (j ∈ S _i ). Using Proposition 6 (1 ⁰ ) we get (i) ∆ ^c _j (G _i ) = r ^c d _j ^c (G _i ) − c ⁽²⁾ _ij , because d _j ^c (G _i ) = r ^c − c. Making use of (i) and (6), we arrive at

∆ ^c _j (G i ) −  τ ^c − θ ^c c − c + r ^c 

d _j ^c (G i ) + (τ ^c − θ ^c ) r ^c − (c τ ^c − c θ ^c )

c − c = 0 . (7)

Case 2. (j ∈ T _i ). Using Proposition 6 (2 ⁰ ) we get (ii) ∆ ^c _j (G _i ) = r ^c d _j ^c (G _i ) − c ⁽²⁾ _ij , because d _j ^c (G _i ) = r ^c − c. Making use of (ii) and (6), we arrive at

∆ ^c _j (G i ) −  τ ^c − θ ^c c − c + r ^c 

d _j ^c (G i ) + (τ ^c − θ ^c ) r ^c − (c τ ^c − c θ ^c )

c − c = 0 . (8)

Finally, using (7) and (8) we find that ∆ ^c _j (G _i ) + p ^c d _j ^c (G _i ) + q ^c = 0 for j ∈ V (G _i ), where p ^c = −( ^τ

_c−c ^−θ

+ r ^c ) and q ^c = ^(τ

^−θ

^{) r}

_c−c ^{−(c τ}

^{−c θ}

⁾ . Since p ^c = − (µ ^c ₁ + µ ^c ₂ ) and q ^c = µ ^c ₁ µ ^c ₂ , we easily obtain the statement ⁵ .  Corollary 1. Let G be a connected or disconnected strongly regular graph of order n and degree r. Then for any vertex deleted subgraph G i we have

n 1,2 = n − 1

2 ± n − 1

r + θ − τ
− 2r 2

q

τ − θ − r
²

− 4 θ ,

3 We say that µ

∈ σ

(G) is the conjugate main eigenvalue if and only if hj, P

ji = n cos

γ > 0, where P

is the orthogonal projection of the space R

onto the eigenspace E

(µ

). The value β

= | cos γ| is called the conjugate main angle of µ

. We know that |M

(G)| = |M(G)| where M(G) and M

(G) denote the set of all main and conjugate main eigenvalues, respectively.

4 If we assume that c

= τ

then using (5) we get c

= −

d

(G

) +

(r

− c) + τ

,

which is reduced to (6). Therefore, without loss of generality we can assume that c

= θ

.

5 In the proof of Theorem 3 it was demonstrated that p = − (µ

+ µ

) and q = µ

µ

. Using the

same arguments one can easily see that p

= − (µ

+ µ

) and q

= µ

µ

.

where n 1 = nβ ₁ ² and n 2 = nβ ₂ ² .

Proof. Since n ₁ + n ₂ = n − 1 and n ₁ µ ₁ + n ₂ µ ₂ = nr − 2r, using (1) we obtain the

statement. 

Corollary 2. Let G be a connected or disconnected strongly regular graph of order n and degree r. Then for any vertex deleted subgraph G _i we have

n ^c _1,2 = n − 1

2 ± n − 1

(c − c) r ^c + θ ^c − τ ^c
− 2(c − c) r ^c 2

q

τ ^c − θ ^c − (c − c) r ^c
2

+ 4(c − c) c τ ^c − c θ ^c
,

where n ^c ₁ = n(β ₁ ^c ) ² and n ^c ₂ = n(β ₂ ^c ) ² . Proof. First, since n ^c ₁ µ ^c ₁ +n ^c ₂ µ ^c ₂ = P

j∈S

d _j ^c (G _i )+ P

j∈T

d _j ^c (G _i ) and r = ^r

^−(n−1)c _c−c , using Proposition 5 we find that n ^c ₁ µ ^c ₁ + n ^c ₂ µ ^c ₂ = (n − 2)r ^c . In view of this and n ^c ₁ + n ^c ₂ = n − 1, we obtain the statement using Theorem 4.  Corollary 3. Let G be a strongly regular graph of order 4n + 1 and degree r = 2n.

If G is a conference graph then µ ^c _1,2 = (2n − 1)a ±

q

4n ² a ² + mb ² and n ^c _1,2 = 2n ± 4n ² a p 4n ² a ² + mb ²

. Corollary 4. Let G be a connected or disconnected strongly regular graph of or- der n and degree r. Then

(A ^c ) ² =  τ ^c − θ ^c c − c



A ^c −  c τ ^c − c θ ^c c − c

 J + 

c ⁽²⁾ _ii + c τ ^c − c θ ^c c − c



I , (9)

where c ⁽²⁾ _ii = (c + c) r ^c − (n − 1)c c.

Proof. It is not difficult to see that relation (9) is true. Since c ⁽²⁾ _ii = rc ² + ((n − 1) − r)c ² and r = ^r

^−(n−1)c _c−c we obtain the assertion.  Definition 2. We say that a simple graph G of order n is conjugate integral if its conjugate eigenvalues are in the form λ ^c _i = p _i + q _i √

m for i = 1, 2, . . ., n, where p _i and q i are integral values. The eigenvalue λ ^c _i = p i + q i

√ m is called the conjugate integral eigenvalue.

Theorem 5 (Lepovi´ c [3]). If a graph G of order n is a conjugate integral graph then its complement G is also conjugate integral .

Proposition 8 (Lepovi´ c [3]). If G is a regular integral graph of order n and degree r then it is conjugate integral too.

Proposition 9 (Lepovi´ c [3]). Let λ ^c ∈ σ ^c (G) be a conjugate eigenvalue of the graph G with multiplicity p ≥ 1 and let q be the multiplicity of the eigenvalue

−λ ^c − 2a ∈ σ ^c (G). Then p − 1 ≤ q ≤ p + 1.

Proposition 10 (Lepovi´ c [3]). Let λ ^c ∈ σ ^c (G) be a conjugate eigenvalue of the graph G with multiplicity p ≥ 1 and let q be the multiplicity of the eigenvalue

λ

+c 2b √

m ∈ σ(G). Then p − 1 ≤ q ≤ p + 1.

In order to obtain some new information’s on strongly regular graphs with respect

to its conjugate adjacency matrices, we need some results on conjugate main and

non-main eigenvalues, as follows.

Proposition 11. Let G be a conjugate integral graph of order n. Let z i = x i +

√ m y i denote the eigenvector of the conjugate integral eigenvalue λ ^c _i = p i + √ m q i , where

x _i = (x ⁽ⁱ⁾ ₁ , x ⁽ⁱ⁾ ₂ , . . . , x ⁽ⁱ⁾ _n ) and y _i = (y ₁ ⁽ⁱ⁾ , y ₂ ⁽ⁱ⁾ , . . . , y ⁽ⁱ⁾ _n ) , and x ⁽ⁱ⁾ _j and y _j ⁽ⁱ⁾ are integers for j = 1, 2, . . ., n. Then z i = x i − √

m y i is the eigenvector of the conjugate integral eigenvalue λ ^c _i = p i − √

m q i ∈ σ ^c (G).

Proof. We know that if λ ^c _i = p i + √

m q i ∈ σ ^c (G) then λ ^c _i = p i − √

m q i ∈ σ ^c (G).

Using the equation λ ^c _i (x ⁽ⁱ⁾ _j + √

m y ⁽ⁱ⁾ _j ) =

n

P

k=1

c jk (x ⁽ⁱ⁾ _k + √

m y _k ⁽ⁱ⁾ ), and keeping in mind that x ⁽ⁱ⁾ _j and y ⁽ⁱ⁾ _j are integers, we get

λ ^c _i (x ⁽ⁱ⁾ _j + √

m y _j ⁽ⁱ⁾ ) =

n

X

k=1

c _jk (G) (x ⁽ⁱ⁾ _k + √

m y ⁽ⁱ⁾ _k ) =

n

X

k=1

c _jk (G) (x ⁽ⁱ⁾ _k − √ m y _k ⁽ⁱ⁾ ) ,

which completes the proof. 

Proposition 12. Let λ ^c _i = p i + √

m q i be the conjugate main eigenvalue of G. Then λ ^c _i = p i − √

m q i is the conjugate main eigenvalue of G.

Proof. Let z i ∈ E A

(λ ^c _i ) so that  

P n

k=1 (x ⁽ⁱ⁾ _k + √

m y _k ⁽ⁱ⁾ ) 

 > 0. Since √

m is an irrational number

 (x ⁽ⁱ⁾ ₁ + x ⁽ⁱ⁾ ₂ + · · · + x ⁽ⁱ⁾ _n ) − √

m (y ₁ ⁽ⁱ⁾ + y ₂ ⁽ⁱ⁾ + · · · + y ⁽ⁱ⁾ _n )   > 0 ,

which completes the proof. 

Corollary 5. Let λ ^c _i = p i + √

m q i be the conjugate non-main eigenvalue of G.

Then λ ^c _i = p i − √

m q i is the conjugate non-main eigenvalue of G.

Proposition 13. Let λ ^c _i = p _i + √

m q _i be the conjugate main eigenvalue of G. Then p _i 6= − a.

Proof. Let us assume, contrary to the statement, that p _i = − a. Then λ _i =

− a + √

m q _i and − λ _i − 2a = − a − √

m q _i have the same multiplicities, which provides that λ ^c _i 6∈ M ^c (G) and − λ ^c _i − 2a 6∈ M ^c (G), a contradiction.  Corollary 6. Let p _i + √

m q _i ∈ σ ^c (G). If p _i = − a then p _i + √

m q _i is the conjugate non-main eigenvalue of G.

Proposition 14. If G is a connected or disconnected strongly regular graph of order n and degree r then it has exactly three distinct conjugate eigenvalues.

Proof. Assume that G is a connected integral ⁶ strongly regular graph. Let λ ₁ = r, λ ₂ and λ ₃ be its distinct eigenvalues with multiplicities m ₁ = 1, m ₂ and m ₃ , respectively. Since G is a conjugate integral graph using Proposition 10 it follows that r ^c = p + q √

m, λ ^c ₂ = − a + (2λ ₂ + 1)b √

m and λ ^c ₃ = − a + (2λ ₃ + 1)b √ m

6 We know that if a strongly regular graph G is not a conference graph then it is integral. In the case that G is a conference graph of order 4n + 1 and degree r = 2n we obtain that r

= 4na and λ

= − a ± bp(4n + 1)m. In fact, in this case G has exactly two distinct conjugate eigenvalues if a = ±b and m = 4n + 1, provided 4n + 1 is not a perfect square. Of course, if √

m = √

2 then

any conference graph has exactly three distinct conjugate eigenvalues.

are distinct conjugate integral eigenvalues of G with multiplicities m 1 , m 2 and m 3 , respectively, where p = (n−1)a and q = − ((n−1)−2r)b. Since G is a regular graph with only one conjugate main eigenvalue r ^c with multiplicity 1 and G is conjugate integral, we obtain from Propositions 9 and 12 that its complement G (possibly disconnected) also has exactly three distinct eigenvalues r ^c = p − q √

m, λ ₂ ^c =

− λ ^c ₂ − 2a and λ ₃ ^c = − λ ^c ₃ − 2a with multiplicities m 1 , m 2 and m 3 , respectively.  Theorem 6. A connected regular graph G of order n and degree r is strongly regular if and only if it has exactly three distinct conjugate eigenvalues.

Proof. In view of Proposition 14 it is sufficient to demonstrate that G is strongly regular if it has exactly three distinct conjugate eigenvalues. Let r ^c , λ ^c ₂ and λ ^c ₃ be the distinct conjugate eigenvalues of G with multiplicities m ₁ , m ₂ and m ₃ , respectively. We know that if λ ^c ∈ σ ^c (G) r M ^c (G) then _2b ^λ

^{√ +c} _m ∈ σ(G) r M(G).

Since |σ ^c (G) r M ^c (G)| = n − 1, λ ₂ = _2b ^λ

^{√ +c} _m and λ ₃ = _2b ^λ

^{√ +c} _m , it follows that G has at least two distinct non-main eigenvalues λ ₂ and λ ₃ and at most three distinct non- main eigenvalues λ 2 , λ 3 and λ r with multiplicities m 2 , m 3 and m 1 − 1, respectively, where λ _r = _2b ^r

^{√ +c} _m .

Consider first the case when G has exactly two distinct non-main eigenvalues.

Since G is a connected regular graph, λ 1 = r is a simple (main) eigenvalue, which proves that G has exactly three distinct eigenvalues with respect to its ordinary adjacency matrix A. In view of this it turns out that G is strongly regular.

We shall now assume that λ r = x + y √

m is a non-main eigenvalue of G, where x = − (n−1)a−(2r−1)

2 and y = ^(n−1)a _2mb . Since x and y are rational numbers and y 6= 0, λ r = x − y √

m must be the eigenvalue of G with the same multiplicity m 1 −1. Since λ 1 = r is a simple eigenvalue, we obtain 1 + m 2 + m 3 + 2(m 1 − 1) > n because

m 1 ≥ 2, a contradiction. 

Proposition 15. A disconnected regular graph G of order n and degree r is strongly regular if and only if it has exactly three distinct conjugate eigenvalues.

Proof. Since G is a connected graph and P ^c

G (λ) = P _G ^c (λ), we obtain the statement

using Theorem 6. 

Proposition 16. Let G be a connected or disconnected strongly regular graph of order n and degree r. Then

τ ^c = c ⁽²⁾ _ii + λ ^c ₂ λ ^c ₃ + c (λ ^c ₂ + λ ^c ₃ ) and θ ^c = c ⁽²⁾ _ii + λ ^c ₂ λ ^c ₃ + c (λ ^c ₂ + λ ^c ₃ ) , where c ⁽²⁾ _ii = (c + c )r ^c − (n − 1)c c.

Proof. First, using Proposition 10 we have λ ₂ = ^λ _c−c

^+c and λ ₃ = ^λ _c−c

^+c . Since r = ^r

^−(n−1)c _c−c , τ = r + λ ₂ λ ₃ + λ ₂ + λ ₃ and θ = r + λ ₂ λ ₃ (see [1]), making use of

(3) and (4) we obtain the statement. 

Proposition 17. Let G be a connected or disconnected strongly regular graph of order n and degree r. Then

λ ^c _2,3 = τ ^c − θ ^c ± q

τ ^c − θ ^c
2

+ 4(c − c ) (c − c )c ⁽²⁾ _ii + c τ ^c − cθ ^c

2 c − c
,

where c ⁽²⁾ _ii = (c + c )r ^c − (n − 1)c c.

Proof. Using Proposition 16 we obtain λ ^c ₂ + λ ^c ₃ = ^τ

_c−c ^−θ

and λ ^c ₂ λ ^c ₃ = − (c ⁽²⁾ _ii +

c τ

−c θ

c−c ), which provides the proof. 

Proposition 18. Let G be a connected or disconnected strongly regular graph of order n and degree r. Then

m 2,3 = n − 1

2 ± (n − 1) θ ^c − τ ^c
− 2(c − c )r ^c 2

q

τ ^c − θ ^c
2

+ 4(c − c ) (c − c )c ⁽²⁾ _ii + c τ ^c − cθ ^c
,

where m 2 and m 3 denote the multiplicities of λ ^c ₂ and λ ^c ₃ , respectively.

Proof. Since m 2 + m 3 = n − 1 and r ^c + m 2 λ ^c ₂ + m 3 λ ^c ₃ = 0, by a straight-forward

calculation we obtain the statement. 

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[2] E. M. Hagos, Some results on graph spectra, Linear Algebra Appl., 356 (2002), 103-111.

[3] M. Lepovi´ c, On conjugate adjacency matrices of a graph, Discrete Math., 307 (2007), 730-738.

[4] M. Lepovi´ c, Some results on walk regular and strongly regular graphs, New Zealand Journal of Mathematics, 38, (2008), 161-169.

Mirko Lepovi´ c

Tihomira Vuksanovi´ ca 32, 34000, Kragujevac, Serbia.

lepovic@kg.ac.rs