Univ. Beograd. Publ. Elektrotehn. Fak.
Ser. Mat. 12 (2001), 68–84.
SOME RESULTS ON GRAPHS WITH
EXACTLY TWO MAIN EIGENVALUES
Mirko Lepović
Let G be a simple graph of order n with exactly two main eigenvalues µ1 and
µ2 . Let β1 and β2 denote the main angles of µ1 and µ2 , respectively. We
show that
µ1,2
n − 2 − µ1 − µ 2
=
±
2
p
(µ1 − µ2 + n)2 − 4n1 (µ1 − µ2 )
,
2
where µ1 and µ2 are the main eigenvalues of the complementary graph G,
and n1 = nβ12 . Besides, we obtain that
n1,2 =
n2 + (n − 2n1 )(µ1 − µ2 )
n
± p
,
2
2 (µ1 − µ2 + n)2 − 4n1 (µ1 − µ2 )
2
where ni = nβ i (i = 1, 2); β 1 and β 2 denote the main angles of µ1 and
µ2 , respectively. Further, let G be any connected or disconnected graph
(not necessarily with two main eigenvalues). Let S be any subset of the
vertex set V (G) and let GS be the graph obtained from the graph G by
adding a new vertex x which is adjacent exactly to the vertices from S.
If σ(GS1 ) = σ(GS2 ) then we prove that σ(GS1 ) = σ(GS2 ) if and only if
σ(GT1 ) = σ(GT2 ), where σ(G) is the spectrum of G and Ti = V (G) \ Si
for i = 1, 2. However, if G is a connected graph which has exactly two
main eigenvalues we prove the following results: (i) if σ(GS1 ) = σ(GS2 )
then σ(GS1 ) = σ(GS2 ); (ii) if σ(Gi ) = σ(Gj ) then σ(Gi ) = σ(Gj ), where
Gk = G \ k denotes the corresponding vertex deleted subgraph of G.
Let G be a simple graph of order n with vertex set V (G) = {1, . . . , n} and
let λ1 ≥ λ2 ≥ . . . ≥ λn be its eigenvalues with respect to its ordinary adjacency
matrix A = A(G). The spectrum of G is the set of its eigenvalues and is denoted
by σ(G). Let PG (λ) = |λI − A| denote the characteristic polynomial of G.
2000 Mathematics Subject Classification: 05 C 50
Keywords and Phrases : Graph, eigenvalue, main eigenvalue
68
Some results on graphs with exactly two main eigenvalues
69
Let µ1 > µ2 > . . . > µm be the distinct eigenvalues of a graph G and let
EA (µi ) denote the eigenspace of the eigenvalue µi (i = 1, . . . , m). We say that
an eigenvalue µ of G is the main eigenvalue if the cosine of the angle between
the eigenspace EA (µ) and the main vector j (whose all coordinates are equal to
1) is different from zero. In other words, the eigenvalue µ is main if and only if
hj, Pji = n cos2 α 6= 0, where P is the orthogonal projection of the space Rn onto
EA (µ). The quantity β = | cos α| is called the main angle of µ.
(k)
Further, let Ak = [aij ] for any non-negative integer k. The number Nk of all
walks of length k in G equals sum Ak , where sum M is the sum of all elements in
a matrix M . According to [2], [3], the generating function HG (t) of the numbers
+∞
P
Nk of walks of length k in the graph G reads HG (t) =
Nk tk . Besides, it was
k=0
proved in [2] that
(1)
HG (t) =
1
t
(−1)n P G (−(t + 1)/t)
−1 ,
PG (1/t)
where G denotes the complement of G. We also note that HG (t) can be represented
in the form
n1 λ
n2 λ
nk λ
,
+
+ ··· +
HG ∗ 1/λ) =
λ − µ1
λ − µ2
λ − µk
where ni = nβi2 and n1 + n2 + · · · + nk = n; µi and βi (i = 1, 2, . . . , k) stand for the
main eigenvalues and main angles of G, respectively. With regard to this notation,
Nm = n1 µ1m + n2 µ2m + · · · + nk µkm for any non-negative integer m.
Let M(G) = {µ1 , µ2 , . . . , µk } be the set of all main eigenvalues of a graph G
of order n. As is known, if λ ∈ σ(G) \ M(G) then − λ − 1 ∈ σ( G ) \ M(G), which
provides the following relation
(3)
k
Y
k
k
X
Y
λ − µm =
λ + µm + 1
1−
m=1
m=1
nm
,
λ + µm + 1
m=1
where µm ∈ M(G) for m = 1, 2, . . . , k.
Theorem 1. Let G be a graph of order n with two main eigenvalues µ1 and µ2 .
Then
p
(µ1 − µ2 + n)2 − 4n1 (µ1 − µ2 )
n − 2 − µ1 − µ2
(4)
µ1,2 =
±
.
2
2
Besides, we have
(5)
n1,2 =
n
n2 + (n − 2n1 )(µ1 − µ2 )
± p
,
2
2 (µ1 − µ2 + n)2 − 4n1 (µ1 − µ2 )
where ni = nβ i (i = 1, 2); β 1 and β 2 denote the main angles of µ1 and µ2 , respectively.
70
Mirko Lepović
Proof. Using (3) by an easy calculation we have (i) µ1 + µ2 = n − 2 − µ1 − µ2 ; and
(ii) µ1 µ2 = µ1 µ2 − (n2 − 1) µ1 − (n1 − 1) µ2 − (n − 1), which provides relation (4).
n
In order to prove relation (5) first recall that e(G)+e(G) =
, where e(G) is
2
n
the number of edges of G. Making use of (i) (n1 µ1 +n2 µ2 )+(n1 µ1 +n2 µ2 ) = 2
;
2
and (ii) n1 + n2 = n, we easily obtain (5).
u
t
Let S be any (possibly empty) subset of the vertex set V (G) and let GS be
the graph obtained from the graph G by adding a new vertex x (x ∈
/ V (G)), which
is adjacent exactly to the vertices from S.
For a square matrix M denote by
P {M }Pthe adjoint of M and for any two
subsets X, Y ⊆ V (G) define hX, Y i = i∈X j∈Y Aij , where A = [Aij ] = {λI −
A}. According to [4], [6], [7], the expression hX, Y i is called the formal product, of
the sets X and Y , associated with the graph G.
For any two disjoint subsets X, Y ⊆ V (G) let X + Y denote the union of
X and Y . It is clear that hX + Y, Zi = hX, Zi + hY, Zi for any Z ⊆ V (G) and
hX, Y i = hY, Xi for any (not necessarily disjoint) X, Y ⊆ V (G). According to [4],
1 1
1
PG (λ)
and hS, Si =
FS
,
(6)
PGS (λ) = PG (λ) λ − FS
λ
λ
λ
λ
where FS (t) =
+∞
P
d(k) tk and d(k) =
P P
i∈S j∈S
k=0
(k)
aij (k = 0, 1, 2, . . . ). The function
FS (t) is called the formal generating function associated with the graph GS . More
generally, we proved in [5] that
(7)
where FX,Y (t) =
hX, Y i =
+∞
P
k=0
PG (λ)
1
FX,Y
λ
λ
e(k) tk and e(k) =
P P
i∈X j∈Y
(X, Y ⊆ V (G)) ,
(k)
aij (k = 0, 1, 2, . . . ). In particular,
setting S • = V (G), we obtain that hS • , S • i = sum {λI − A} and FS • (t) = HG (t).
We also note from (6) that for any S ⊆ V (G),
(8)
PGS (λ) = λPG (λ) − hS, Si ,
where hS, Si is the formal product associated with G.
Let i be a fixed vertex from the vertex set V (G) and let Gi = G \ i be its
corresponding vertex deleted subgraph.
Proposition 1 (Lepović [8]). Let G be a connected or disconnected regular graph
of order n and degree r. Then for any i ∈ V (G) and any S ⊆ V (G) we have :
(−1)n−1 PG ( λ ) (1◦ )
P Gi (λ) =
λ − r PGi ( λ ) −
;
λ+r+1
λ+r+1
n − 2 |S|
(2◦ )
PGT (λ) = PGS (λ) −
PG (λ) ;
λ−r
71
Some results on graphs with exactly two main eigenvalues
2
λ + r + 1 − |S|
(−1)n+1 P GS (λ) =
λ − r PGS ( λ ) +
PG ( λ ) ,
λ+r+1
λ+r+1
◦
(3 )
where r = (n − 1) − r, λ = − λ − 1 and T = V (G) \ S.
Let G be the union of any k (not necessarily connected) graphs G(1) , G(2) ,
. . . , G and let S = S1 ∪ S2 ∪ . . . ∪ Sk ⊆ V (G), where Si ⊆ V (G(i) ) for i = 1, 2,
. . . , k. In [6] it was proved that
(k)
(9)
PGS (λ) =
k X
PG(i) (λ)
Si
i=1
Y
k
Y
PG(j) (λ) − (k − 1) λ
PG(i) (λ) ,
j∈Vi
i=1
where Vi = {1, 2, . . . , k} \ {i}. Using (9) and Proposition 1 (2◦ ), we easily obtain
the following result.
Proposition 2. Let G be the union of k regular graphs G1 , G2 , . . . , Gk of order
k
S
n1 , n2 , . . . , nk and degree r1 , r2 , . . . , rk , respectively. Then for any S =
Sm ⊆
m=1
V (G), we have
(10)
k
X
nm − 2 |Sm | PGT (λ) = PGS (λ) −
PG (λ) ,
λ − rm
m=1
where T = V (G) \ S and Sm ⊆ V (Gm ) for m = 1, 2, . . . , k.
Proposition 3. Let G be the union of k regular graphs G1 , G2 , . . . , Gk of order
n1 , n2 , . . . , nk and degree r1 , r2 , . . . , rk , respectively. Then for any vertex i ∈ V (G` ),
we have
k
X
(−1)n−1 P Gi (λ) = 1 −
nm
PG ( λ )
PGi ( λ ) −
2 ,
λ
+
r
+
1
m
λ + r` + 1
m=1
where λ = − λ − 1 and ` = 1, 2, . . . , k.
Q
Proof. Let i ∈ V (G` ) be a fixed vertex of G. Since PGi (λ) = PGi` (λ)
PGm (λ)
m∈V`
P
and HGi (t) = HGi` (t) +
HGm (t), making use of relation (1) and Proposition 1
m∈V`
(1◦ ), an easy calculation yields the statement.
u
t
Proposition 4. Let G be the union of k regular graphs G1 , G2 , . . . , Gk of order n1 , n2 , . . . , nk and degree r1 , r2 , . . . , rk , respectively. If PGi (λ) = PGj (λ) then
P Gi (λ) = P Gj (λ).
Proof. Since e(Gk ) = e(G) − deg(k) and e(Gi ) = e(Gj ) it follows that both
vertices i, j belong to the same component G` . Consequently, using Proposition 3
we obtain at once P Gi (λ) = P Gj (λ).
72
Mirko Lepović
Proposition 5 (Lepović [5,6,7]). Let G be any graph of order n and let S ⊆ V (G).
Then
PGS (λ) − PGT (λ) = (−1)n P GS (− λ − 1) − P GT (− λ − 1) ,
where T = V (G) \ S.
Corollary 1. Let G be the union of k regular graphs G1 , G2 , . . . , Gk of order n1 , n2 ,
k
S
. . . , nk and degree r1 , r2 , . . . , rk , respectively. Then for any S =
Sm ⊆ V (G),
m=1
we have
n+1
(−1)
P
n+1
GT
(λ) = (−1)
k
X
n m − 2 Sm P GS (λ) −
PG ( λ ) ,
λ + rm + 1
m=1
where λ = − λ − 1 and T = V (G) \ S, Sm ⊆ V (Gm ) for m = 1, 2, . . . , k.
Proposition 6. Let G be the union of k regular graphs G1 , G2 , . . . , Gk of order
k
S
n1 , n2 , . . . , nk and degree r1 , r2 , . . . , rk , respectively. Then for any S =
Sm ⊆
m=1
V (G), we have
n+1
(−1)
k
X
k
X
Sm 2
nm
P GS (λ) = 1 −
PGS ( λ ) +
PG ( λ )
λ + rm + 1
λ + rm + 1
m=1
m=1
k
X
2 Sm + 1−
PG ( λ ) ,
λ + rm + 1
m=1
where λ = − λ − 1 and T = V (G) \ S, Sm ⊆ V (Gm ) for m = 1, 2, . . . , k.
Proof. We shall prove the statement by induction on k. It is obviously true for
k = 1. Assume that k > 1 and assume that Proposition (6) is satisfied for any
i < k.
First, we note that P GS (λ) = λ P G (λ) − hT, T i, where hT, T i is the formal
product associated with G, T = T1 ∪ T2 ∪ . . . ∪ Tk and Tm = V (Gm ) \ Sm for
m = 1, 2, . . . , k. Let Hi = G \ Gi and let Ri = S \ Si for i = 1, 2, . . . , k. Since
HGRi (t) = H(Hi )Ri (t)+HGi (t), using the induction hypothesis for the graph (Hi )Ri ,
ni
the fact that P Gi (λ) = 1−
(−1)ni PGi ( λ ) and (1), an easy calculation
λ + ri + 1
yields that
n+1
(−1)
P
GRi
k
X
X
Sm 2
nm
(λ) = 1 −
PGRi ( λ ) +
PG ( λ )
λ + rm + 1
λ + rm + 1
m=1
m∈Vi
k
X
2 Sm + 1−
PG ( λ ) ,
λ + rm + 1
m∈Vi
73
Some results on graphs with exactly two main eigenvalues
(λ) − hRi , Ri i,
where Vi = {1, 2, . . . , k} \ {i}. On the other hand, P
(λ) = λP
where Ri = T1 ∪ · · · ∪ Ti−1 ∪ Ti• ∪ Ti+1 ∪ · · · ∪ Tk and
we have Ri = T ∪ Si . Therefore,
V (Gi ). Since Ti• = Si ∪ Ti
(11)
P
GRi
(λ) = λ P
G
G Ri
Ti• =
G
(λ) − hT, T i − hSi , Si i − 2 hSi , T i ,
where hX, Y i is the formal product associated with G. Further, we find from
(8) that hSi , Si i = λ P G (λ) − P GT (λ), where Ti = V (G) \ Si . According to
i
Proposition (5), note that P GT (λ) = P GS (λ) + (−1)n (PGTi ( λ ) − PGSi ( λ )).
i
Pi
Since HGSi (t) = H(Gi )Si (t) + m∈Vi HGm (t), using the induction hypothesis for
the graph (Gi )Si and equations (1) and (10), we get
(−1)n+1 P
GTi
k
X
(λ) = 1 −
nm
PGSi ( λ ) + PG (λ )
λ + rm + 1
m=1
S 2
i
PG ( λ ) .
+
λ + ri + 1
Combining (11) with the last relation and the first equation of Proposition
6, we arrive at
n i − 2 Si n+1
n+1
(−1)
P GS (λ) = (−1)
λ P G (λ) + 2 hSi , T i +
P G( λ )
λ + ri + 1
k
X
nm
+ 1−
PGRi ( λ ) − PGSi ( λ )
λ + rm + 1
m=1
X
S 2 Sm 2
i
+
−
PG ( λ )
λ + rm + 1
λ + ri + 1
m∈Vi
X
n m − 2 S m n i − 2 Si +
−
PG ( λ )
λ + rm + 1
λ + ri + 1
m∈Vi
ni
+
PG ( λ ) .
λ + ri + 1
Using now relations:
Pk ni −2|Si | n+1
(i)
P GS (λ) − P GT (λ) (see Corollary 1);
i=1 λ+ri +1 PG ( λ ) = (−1)
Pk
(ii) i=1 PGRi ( λ ) − PGSi ( λ ) = (k − 2) PGS ( λ ) + (λ + 1) PG ( λ ) (see (9));
Pk
Pk P
|Si | 2
|Sm | 2
|Sm | 2
− λ+r
=
(k
−
2)
(iii) i=1
m=1
m∈Vi λ+rm +1
+1
λ+r
i
m +1
(iv)
(it is easy to see);
Pk P
nm −2|Sm | i=1
m∈Vi
and using
λ+rm +1
−
ni −2|Si | λ+ri +1
= (k − 2)
nm −2|Sm | m=1 λ+rm +1 ;
Pk
74
(v)
Mirko Lepović
Pk
ni
i=1 λ+ri +1
PG ( λ ) = PG ( λ ) + (−1)n+1 P
G
(λ)
(see (1) and (2)), the previous relation is transformed into
(−1)n+1 k P
GS
(λ) = (−1)n+1 k λ P
G
(λ) + 2 hS, T i + P
GS
(λ) − P
GT
(λ)
k
X
+ (k − 2) 1 −
nm
PGS ( λ ) + (λ + 1) PG ( λ )
λ + rm + 1
m=1
X
k
k
X
Sm 2
n m − 2 Sm + (k − 2)
+
PG ( λ )
λ + rm + 1
λ + rm + 1
m=1
m=1
+ PG ( λ ) + (−1)n+1 P G (λ) .
Next, setting S • = V (G) we have hS • , S • i = hS, Si + 2 hS, T i + hT, T i. Then
using (8) we find that P GS (λ) + P GT (λ) = 2 λP G (λ) − hS • , S • i + 2 hS, T i. Since
FS • (t) = H G (t), according to (1) and (7) we arrive at
2 hS, T i = P
GS
(λ) + P
GT
(λ) − (2λ + 1) P
G
(λ) + (−1)n PG ( λ ) .
Finally, combining the last two relations with (v) we obtain that Proposition
6 is true for any k.
u
t
Corollary 2. Let G be the union of k regular graphs G1 , G2 , . . . , Gk of order
k
S
n1 , n2 , . . . , nk and degree r1 , r2 , . . . , rk , respectively. Then for any S =
Sm ⊆
m=1
V (G), we have
(−1)n+1 P
GT
k
X
(λ) = 1 −
nm
PGS ( λ ) + PG ( λ )
λ + rm + 1
m=1
k
X
Sm 2
+
PG ( λ ) ,
λ + rm + 1
m=1
where λ = − λ − 1 and T = V (G) \ S, Sm ⊆ V (Gm ) for m = 1, 2, . . . , k.
Let G be a connected or disconnected graph of order n with k main eigenvalues µ1 , µ2 , . . . , µk . Let µ ∈ M(G) and let G be any regular graph of ’order’
n β 2 and ’ degree’ µ, where β denotes the main angle of µ. In this work G is
called the µ-regular graph. Let G1 , G2 , . . . , Gk denote the µ-regular graphs of or2
der n1 , n2 , . . . , nm and degree µ1 , µ2 , . . . , µm , respectively, where nm = nβm
for
1
m = 1, 2, . . . , k, such that G = G1 ∪ G2 ∪ · · · ∪ Gk , understanding that (1• )
1 Note that V (G) = V (G ) ∪ V (G ) ∪ · · · ∪ V (G ) is not correct because if x ∈ V (G) then
1
2
k
the whole vertex x not necessarily belongs to some component Gm . Namely, if x ∈ V (G) then
we understand that one piece of the vertex x belongs to Gm1 , some other piece of x belongs to
Gm2 , . . . , and the last one belongs to Gmi , where 1 ≤ m1 < m2 < · · · < mi ≤ k (1 ≤ i ≤ k).
75
Some results on graphs with exactly two main eigenvalues
V (Gi )∩V (Gj ) = ∅ whenever i 6= j (i, j = 1, 2, . . . , k); and 2 (2• )
P
j∈V`
m
(m,`)
aij
= µm
`
(m,`)
for any non-negative integer m, where i, j ∈ V` = V (G` ) and A (G` ) = [aij
for ` = 1, 2, . . . , k.
]
nm λ
We note that relation (2• ) is consistent with HGm ( λ1 ) = λ−µ
, and according
m
to (1 ), HG (t) = HG1 (t) + HG2 (t) + · · · + HGk (t) is consistent with relation (2).
k
S
Since G = G1 ∪G2 ∪· · ·∪Gk it follows that GS is the graph with S =
Sm
•
m=1
so that |S| = |S1 | + |S2 | + · · · + |Sk |, where Sm ⊆ V (Sm ) for m = 1, 2, . . . , k. In
view of this fact it is easy to see that all previous results concerning GS are also
true for GS which is related to any graph G with k main eigenvalues.3
In what follows, for the sake of brevity, let PS and PS denote PGS (λ) and
P GS (− λ − 1), respectively, understanding that P = PG (λ) and P = P G (−λ − 1).
Proposition 7. For any graph G of order n and any S ⊆ V (G), we have
2
(12)
(−1)n+1 4 PS P + P PT + P P = PS − PT + (−1)n P − P ,
where T = V (G) \ S.
Proof. Let µ1 , µ2 , . . . , µk denote the main eigenvalues of G and let S ⊆ V (G),
having in mind that S = S1 ∪ S2 ∪ · · · ∪ Sk , where Sm ⊆ V (Gm ) for m = 1, 2, . . . , k.
Using (1), (2) and (10) we obtain that
k
X
Sm (13)
2
P = PT − PS + (−1)n P − P .
λ
−
µ
m
m=1
Combining Corollary 2 with the last relation, by a straightforward calculation
we arrive at
2
(−1)n+1 4 PT P + P PS + P P = PT − PS + (−1)n P − P ,
u
t
which provides the proof.
Theorem 2. Let G be a graph of order n and let σ(GS1 ) = σ(GS2 ). Then their
complementary graphs GS1 and GS2 are cospectral if and only if GT1 and GT2 are
cospectral, where Tm = V (G) \ Sm for m = 1, 2.
(m,`)
2 Relation (2• ) may be explained as follows. Namely, a
can be treated as the number
ij
of walks of length m starting at ’ the vertex’ i and terminating at ’ vertex’ j. According to this
P
(m,`)
fact,
a
is the number of walks of length m starting at the vertex i and terminating
j∈V` ij
anywhere. Since from any arbitrary vertex j, a walk can be continued in exactly µ` ways, we
P
(m,`)
obtain that
a
= µm
` .
j∈V` ij
3 In order to prove any result which is related to G we shall consider G
m as a ’ normal’ regular
S
graph. For the sake of an example, PGm (λ) denotes ’ the characteristic polynomial’ of Gm and
PG (λ) =
Qk
m=1
PGm (λ) represents the characteristic polynomial of G. Thus, in the figurative
nm λ
and relation
λ−µm
nm
n
m
(−1) PGm (λ).
λ+µm +1
1
sense, according to HGm ( λ
)=
P
Gm
(λ) = 1 −
(1), the characteristic polynomial of Gm reads
76
Mirko Lepović
Proof. First, let PS1 = PS2 and PT1 = PT2 . Using relation (12) we obtain implicitly
PS1 = PS2 . Conversely, suppose that PS1 = PS2 and PS1 = PS2 . According to (12)
we get
(−1)n 4 P (PT1 − PT2 ) = (PT1 − PT2 ) 2 PS1 − (PT1 + PT2 ) + 2 (−1)n P − P .
Let assume, contrary to
the statement, that PT1 6= PT2 . Then the last relation
is reduced to 2 (−1)n P + P = 2PS1 − (PT1 + PT2 ), from which we obtain 4 · λn =
0 · λn , a contradiction.
u
t
Corollary 3. If σ(GS1 ) = σ(GS2 ) then their complementary graphs GS1 and GS2
are cospectral if and only if GT1 and GT2 are cospectral.
Proposition 8. Let G be any graph of order n with k main eigenvalues µ1 , µ2 ,
k
S
(`) (`)
. . . , µk and let S` =
Sm ⊆ V (G), where Sm ⊆ V (Sm ) for ` = 1, 2, m =
m=1
(1)
(2)
1, 2, . . . , k. If σ(GS1 ) = σ(GS2 ) then σ(GT1 ) = σ(GT2 ) if and only if |Sm | = |Sm |
for m = 1, 2, . . . , k, where T` = V (G) \ S` for ` = 1, 2.
(1)
(2)
Proof. If |Sm | = |Sm | for m = 1, 2, . . . , k using (10) it follows that GT1 and GT2
are cospectral. Conversely, assume that GT1 and GT2 are two cospectral graphs.
According to (10), we have
k
k
(2)
(1)
X
X
|Sm |
|Sm |
=
,
λ − µm
λ − µm
m=1
m=1
(1)
(2)
which provides that |Sm | = |Sm | for m = 1, 2, . . . , k.
u
t
Further, using (10) note that
(λ − µ ) P (λ) − P (λ) nm
m
GT
GS
|Sm | =
+ lim
,
λ→µm
2
2 PG (λ)
(14)
for m = 1, 2, . . . , k. The last relation can be also easily obtained by relation (13)
(λ − µm ) (−1)n P G (− λ − 1) using that nm = lim
. Let now µm be a simple
λ→µm
PG (λ)
main eigenvalue of G. Then from Corollary 2 it follows that
lim
λ→µm
(λ − µm ) (−1)n+1 P
GT
(λ) = nm PGS (µm ) + |Sm |2 lim
λ→µm
PG (λ)
,
λ − µm
from which we readily obtain
2
Sm = −nm PGS (µm ) .
PG0 (µm )
(15)
Proposition 9. Let G be a graph with k main eigenvalues µ1 , µ2 , . . . , µk and let
S
k
(`)
(`)
S` =
Sm ⊆ V (G), where Sm ⊆ V (Gm ) for ` = 1, 2. If µ1 , µ2 , . . . , µk
m=1
77
Some results on graphs with exactly two main eigenvalues
(1)
(2)
are simple and GS1 and GS2 are cospectral then |Sm | = ±|Sm | with equality
(1)
(2)
|Sm | = |Sm | for m = 1, 2, . . . , k if and only if GS1 and GS2 are cospectral.
Proof. Using Theorem 2, Proposition 8 and relation (15), we immediately obtain
the statement.
u
t
Let now consider the graph G displayed in Fig. 1 and let S1 = {3, 5} and
S2 = {6, 8}. It easy to see that (i) σ(G) = {± 2.2143, ± 1.6751, ± 0.5392, ± 0}; (ii)
GS1 and GS2 are cospectral without cospectral complements and (iii) all non-zero
eigenvalues of G are main.
2
v
Setting S1 =
S
6
8
7
v
v
v
v
v
v
v
1
3
4
5
6
(1)
Sm
Fig. 1
S
6
(2) (`)
and S2 =
Sm where Sm ⊆ V (Gm ) for
m=1
m=1
` = 1, 2, we have the following data which are related to G in Fig 1.
(1)
2.1362
|S1 | =
(1)
0.3791
|S2 | = − 0.3791
µ1 =
2.2143
n1 = 6.8665
|S1 | =
µ2 =
1.6751
n2 = 0.5616
|S2 | =
µ3 =
0.5392
n3 = 0.3741
|S3 | = − 0.1724
µ4 = − 0.5392
n4 = 0.0169
|S4 | =
µ5 = − 1.6751
n5 = 0.1167
|S5 | = − 0.1728
µ6 = − 2.2143
n6 = 0.0643
|S6 | = − 0.2068
(1)
(1)
0.0366
(2)
2.1362
(2)
(2)
0.1724
(2)
0.0366
|S3 | =
|S4 | =
(1)
|S5 | = − 0.1728
(2)
(1)
|S6 | =
(2)
0.2068
Table 1
Next, let µm be a main eigenvalue of G with multiplicity p ≥ 1 and denote by
(1)
(1)
(1)
(1)
µ1 , µ2 , . . . , µ` the distinct eigenvalues of GS , by understanding that µm = µm
if µm ∈ σ(GS ). Then according to (1), (2) and Proposition 6, we have
`
k
k
(1)
Sm 2 PG (λ)
X
X
X
nm nm 1+
= 1+
+ 1+
,
(1)
λ − µm
λ − µm PGS (λ)
m=1 λ − µm
m=1
m=1
(1)
(1)
2
where nm = (n + 1) βm
(GS ) and βm (GS ) is the main angle of µm for m =
1, 2, . . . , `, keeping in mind that some of them could be zero.
(1)
We note from the last relation that if µm = µm ∈ σ(GS ) has the multiplicity
(i) p or / and (ii) p + 1 then it must be |Sm | = 0. In the case (i) we find easily
78
Mirko Lepović
(1)
(1)
nm = nm . In both cases µm is the main eigenvalue of GS . If |Sm |2 > 0 then the
previous relation provides that
(16)
(λ − µ ) P (λ) m
GS
.
λ→µm
PG (λ)
|Sm |2 = (n(1)
m − nm ) lim
Let G be a graph with only simple main eigenvalues µ1 , µ2 , . . . , µk and let
k
S
(`) S` =
Sm ⊆ V (G) for ` = 1, 2, such that GS1 and GS2 are cospectral with
m=1
(2) (1) non-cospectral complements. Besides, let M± = m Sm = ±Sm . Since
P
P
(`)
(`)
|S1 | = |S2 | and |S` | =
m∈M+ |Sm | +
m∈M− |Sm | we get easily (i) |S` | =
P
P
(`)
(`)
m∈M+ |Sm | and (ii)
m∈M− |Sm | = 0.
Proposition 10. Let G be a graph with simple main eigenvalues µ1 , µ2 , . . . , µk
and let GS1 and GS2 be two cospectral graphs such that their complementary graphs
GS1 and GS2 are not cospectral. If S1 ∩ S2 = ∅ then for any m ∈ M− we have
µm ∈ σ(GS ) and µm 6∈ σ(GT ), where S = S1 ∪ S2 and T = V (G) \ S.
Proof. Let S = S1 ∪ S2 ∪ . . . ∪ Sk where Sm ⊆ V (Gm ) for m = 1, 2, . . . , k. Since
(1)
(2)
(1)
S1 ∩ S2 = ∅ it follows that |Sm | = |Sm | + |Sm |. So we find that |Sm | = 2 |Sm | for
m ∈ M + and |Sm | = 0 for m ∈ M − . Consequently, since |Tm | = nm − |Sm | we
get |Tm | = nm > 0 for any m ∈ M − . Therefore, according to (15) we obtain for
m ∈ M − that µm ∈ σ(GS ) and µm 6∈ σ(GT ).
u
t
Let come back to the graph G displayed in Fig. 1. Namely, if we set S =
{3, 5, 6, 8} then Proposition 10 and Table 1 provide that µ2 = 1.6751, µ3 = 0.5392
and µ6 = − 2.2143 belong to the spectrum of GS . Besides, since µ2 , µ3 and µ6 are
(1)
(1)
(1)
simple eigenvalues of GS we find that n2 = n2 , n3 = n3 and n6 = n6 , which
can be verified easily.
(2)
(2)
(2)
Let now denote by µ1 , µ2 , . . . , µ` the distinct eigenvalues of GT and let
(2)
(2)
2
nm = (n + 1) βm
(GT ), where T = V (G) \ S and βm (GT ) is the main angle of µm .
(2)
Besides, let µm = µm ∈ σ(GT ) and let assume that |Sm | 6∈ {0, nm }. Then we also
have |Tm | 6∈ {0, nm } and, according to (16) we get
(17)
(λ − µ ) P (λ) 2
m
GT
nm − |Sm | = (n(2)
.
m − nm ) lim
λ→µm
PG (λ)
(1)
(2)
Of course, in this case both µm ∈ σ(GS ) and µm ∈ σ(GT ) have the multiplicity p − 1. Combining (14), (16) and the last relation we obtain that
(λ − µ ) P (λ) (λ − µ ) P (λ) m
GS
m
GT
= n(2)
lim
.
m
λ→µm
λ→µm
PG (λ)
PG (λ)
n(1)
lim
m
(1)
(2)
We note from the previous relation that either µm ∈ M(GS ) and µm ∈
(1)
(2)
(1)
M(GT ) or µm 6∈ M(GS ) and µm 6∈ M(GT ). However, if µm ∈ σ(GS ) has
79
Some results on graphs with exactly two main eigenvalues
(1)
the multiplicity p or p + 1 then it is not difficult to see that µm ∈ M(GS ) and
(λ−µm ) PGT (λ) (2)
µm 6∈ M(GT ). Indeed, since in that case nm = − lim
(see (14)),
PG (λ)
λ→µm
using (17) we get
(2)
nm
= 0.
Proposition 11. Let µm be a main eigenvalue of G with multiplicity p ≥ 2 and
(1)
(2)
let |Sm | = 0. Then (i) µm = µm ∈ M(GS ) and (ii) µm = µm ∈ M(GT ) if and
(1)
only if µm is an eigenvalue of GS with multiplicity p − 1.
Let G = H 1 ∪ H 2 ∪ . . . ∪ H k , where H m is the µ-regular graph of order
2
nm = n β m and β m is the main angle of µm ∈ M(G) for m = 1, 2, . . . , k. Let
k
k
S
S
S =
Sm ⊆ V (G) and let S =
Rm ⊆ V (G), by understanding that
m=1
m=1
Sm ⊆ V (G m ) and Rm ⊆ V (H m ). We note from Corollary 1 that
PGT (λ) = PGS (λ) −
k
X
nm − 2 |R m | (−1)n P
λ
+
µ
+
1
m
m=1
G
(λ),
where λ = − λ − 1. Making use of (10) and the last relation by an easy calculation
we arrive at
k
X
k
k
X
X
|Sm |
|R m | nm =
1+
,
λ − µm
λ + µm + 1
λ − µm
m=1
m=1
m=1
from which we easily get
(18)
|Sm | =
k
X
i=1
|R i |
nm
µm + µi + 1
and |R m | =
k
X
i=1
| Si |
nm
µm + µi + 1
for m = 1, 2, . . . , k. In particular, if S = V (G) then |Sm | = nm and |R m | = nm for
m = 1, 2, . . . , k, which provides that
(19)
k
X
i=1
ni
=1
µm + µi + 1
Let S = {i} and let {i} =
k
S
and
k
X
i=1
(i)
Im
ni
= 1.
µm + µi + 1
(i)
for i = 1, 2, . . . , n, where Im ⊆ V (G m ).
m=1
We note that G{i} is the overgraph of G obtained from the graph G by adding
k
n
P
P
(i)
(i)
a pendant edge at vertex i. Clearly, (i)
|Im | = 1 and (ii)
|Im | = nm for
m=1
i=1
i = 1, 2, . . . , n and m = 1, 2, . . . , k.
(i)
(i)
(i)
The numbers |I1 |, |I2 |, . . . , |Ik | represent a decomposition of the vertex
i with respect to G 1 , G 2 , . . . , G k , respectively. Table 2 contains the ’pieces’
(i)
(i)
(i)
|I1 |, |I2 |, . . . , |I6 | for any vertex i of the graph G shown in Fig 1.
80
Mirko Lepović
i→
1
2
3
4
5
6
7
8
|I1 |
(i)
0.3323
0.3323
0.7358
0.9647
1.4004
1.0681
0.9647
1.0681
(i)
|I2 |
0.2808
0.2808
0.4704
0.2263
– 0.0912
– 0.1896
– 0.2263
– 0.1896
|I3 |
(i)
0.1870
0.1870
0.1009
– 0.3197
– 0.2732
0.0862
0.3197
0.0862
(i)
|I4 |
0.0397
0.0397
– 0.0214
– 0.0679
0.0580
0.0183
– 0.0679
0.0183
|I5 |
(i)
0.1280
0.1280
– 0.2144
0.1032
0.0416
– 0.0864
0.1032
– 0.0864
(i)
|I6 |
0.0322
0.0322
– 0.0712
0.0934
– 0.1355
0.1034
– 0.0934
0.1034
Table 2
(1)
Let now turn to the data displayed in Tables 1 and 2. First, for |Sm | and
(1)
(3)
(5)
(2)
(6)
(8)
we have |Sm | = |Im | + |Im | and |Sm | = |Im | + |Im |. We note from Fig. 1
that vertices 1 and 2 are similar under the automorphism group, which provides
(1)
(2)
(4)
(7)
that |Im | = |Im | for m = 1, 2, . . . , 6. We also note from Table 2 that |Im | = ±|Im |
for m = 1, 2, . . . , 6. One can easy to verify that G{4} and G{7} are two cospectral
graphs without cospectral complements.4
(2)
|Sm |
Proposition 12. Let G be any graph of order n with k main eigenvalues µ1 , µ2 ,
. . . , µk . Then for any vertex i ∈ V (G), we have
n−1
(−1)
(i) m
X
2
Im nm
PGi ( λ ) −
PG ( λ ) ,
P Gi (λ) = 1 −
λ + µm + 1
λ + µm + 1
m=1
m=1
k
X
where λ = − λ − 1.
Proof. Since PG{i} (λ) = λ PG (λ) − PGi (λ) using Corollary 2 and relations (1) and
(2), by a straightforward calculation we obtain the statement.
u
t
(0)
(0)
(0)
(0)
Let µ1 , µ2 , . . . , µ` be the distinct eigenvalues of G with mind that µm =
µm ∈ M(G) for m = 1, 2, . . . , k. Let {e1 , e2 , . . . , en } be the standard orthonormal
basis of Rn and let Pi be the orthogonal projection of the space Rn onto EA (µi ).
Then for any graph G (connected or disconnected) and any subset S of the
vertex set V (G), we have [9]
(20)
`
X
ρ2i (S) PGS (λ) = PG (λ) λ −
(0)
i=1 λ − µi
where
X
ρi (S) = Pi e j .
j∈S
4 All data given in Tables 1 and 2 are obtained by using the program called ’ REGULAS’,
which has been written in the programming language C. This program is applicable for any graph
(connected or disconnected) on 2, 3, . . . , 16 vertices.
81
Some results on graphs with exactly two main eigenvalues
(0)
Next, let µm be an eigenvalue of G with multiplicity p ≥ 1 and let denote
(m) (m)
(m)
by { (z1 , z2 , . . . , zn ) m = 1, 2, . . . , p} a complete set of mutually orthogonal
(0)
normalized eigenvectors which correspond to µm such that
n
X
(1)
zi
≥0
and
n
X
i=1
(m)
zi
=0
i=1
(0)
for m = 2, 3, . . . , p. In the case that µm 6∈ M(G) we easily get ρm (S) = ρm (T ),
(0)
where T = V (G) \ S. However, if µm is the main eigenvalue of G then according
to (20) and the last relation we obtain that
(21)
ρ2m (T ) =
√
nm −
X
(1)
zi
2
+
p X
X
m=2
i∈S
(m)
zi
2
,
i∈S
(0)
(m)
(m)
(m)
2
where nm = nβm
and βm is the main angle of µm . Let now (x1 , x2 , . . . , xn )
n
P
√
(m)
denote the eigenvector of µm so that
xi = nm . Using (20) and (21) we get
i=1
P
√
(m) k
X
nm − 2 nm
i∈S xi
PGT (λ) = PGS (λ) −
PG (λ) .
λ − µm
m=1
Since e(GS ) = e(G) + |S| and n1 + n2 + · · · + nk = n it follows from the
k √
P (m) P
nm
xi
. Finally, using (10) we arrive at
previous relation that |S| =
m=1
Proposition 13. For any S =
i∈S
k
S
m=1
P (m) √
Sm ⊆ V (G) we have |Sm | = nm
xi
.
i∈S
Proposition 14. Let µ1 , µ2 , . . . , µk ∈ M(G). Then :
√
√
√
(1)
(2)
(k)
n1 xi + n2 xi + · · · + nk xi = 1 ;
(1◦ )
√
√
√
(1)
(2)
(k)
(2◦ )
n1 xi µ1 + n2 xi µ2 + · · · + nk xi µk = deg(i) ,
for any i = 1, 2, . . . , n.
Proof. Let M = { m | λm ∈ σ(G) \ M(G)}. As is known, for any non-negative
integer `, we have
k
X
X
`
A =
Xm µ`m +
Ym λ`m ,
m=1
m∈M
(m)
(m)
(m)
(m) (m)
where Xm = Xm [xij ] and Ym = Ym [ yij ] with zij = zi zj . Of course, here
(m) (m)
(m)
{ (y1 , y2 , . . . , yn ) m ∈ M } represents a complete set of mutually
orthogonal
normalized eigenvectors which are related to {λm ∈ σ(G) \ M(G) m ∈ M } such
82
Mirko Lepović
(m)
that y1
(22)
(m)
+ y2
n
X
(m)
+ · · · + yn
(`)
aij =
k X
n
X
m=1
j=1
= 0. Using the last relation we get
(m)
xj
(m) `
µm
xi
j=1
+
n
X X
m∈M
(m)
yj
(m) `
λm
yi
,
j=1
u
t
which provides the proof.
Corollary 4. Let G be a connected or disconnected graph of order n with exactly
(1)
2
two main eigenvalues µ1 and µ2 . Then xi = √deg(i)−µ
n1 ( µ1 −µ2 ) for i = 1, 2, . . . , n.
We note from (22) and Proposition 13 that for any S ⊆ V (G) the following
k
P
|Sm | •
relation is satisfied FS,S • λ1 = λ
λ−µm , where S = V (G). Consequently,
m=1
in view of Propositions 6 and 12 and Corollary 2 we obtain the next result.
Proposition 15. Let G be any connected or disconnected graph of order n with
k main eigenvalues µ1 , µ2 , . . . , µk . Then for any i ∈ V (G) and any S ⊆ V (G) we
have :
k
X
(λ) = 1 +
PG (λ) 2
nm 1
PGS (λ) + PG (λ) +
FS,S •
;
2
λ − µm
λ
λ
m=1
(−1)n+1 P
GT
(−1)n+1 P
k
X
(λ)
=
1
+
GS
(−1)n−1 P
k
X
=
1
+
(λ)
Gi
nm 1
1 2
PGS (λ) + 1 + FS,S •
) PG (λ) ;
λ − µm
λ
λ
m=1
nm PG (λ) 2
1
PGi (λ) −
F{i},S •
,
2
λ − µm
λ
λ
m=1
where λ = − λ − 1 and T = V (G) \ S.
(m)
Proposition 16. Let µ1 , µ2 , . . . , µk ∈ M(G) and let (x1
n
P
√
(m)
unit eigenvector of µm so that
xi = nm . Then
(m)
, x2
(m)
, . . . , xn
) be the
i=1
(23)
(m)
xi
=
k
X
j=1
√
(j)
√
nj xi
nm ,
µm + µj + 1
2
for i = 1, 2, . . . , n and m = 1, 2, . . . , k, understanding that nm = n β m where β m is
the main angle of µm .
Proof. Using Proposition 13 and using the right-hand side of relations (18) and
(19), we obtain the required statement.
u
t
(m)
(m)
(m)
Using (23) and keeping in mind that {(x1 , x2 , . . . , xn ) m = 1, 2, . . . , k}
is the complete system of mutually orthogonal normalized eigenvectors, we arrive
at
83
Some results on graphs with exactly two main eigenvalues
Proposition 17. Let G be a graph with k main eigenvalues. Then for any m =
1, 2, . . . , k we have
k
X
1
ni
=
2
nm
i=1 µm + µi + 1
k
and
X
1
ni
=
2 .
nm
i=1 µm + µi + 1
Theorem 3. Let G be a connected graph with exactly two main eigenvalues µ1
and µ2 and let σ(GS1 ) = σ(GS2 ). Then (i) σ(GS1 ) = σ(GS2 ); and (ii) σ(GT1 ) =
σ(GT2 ), where Tm = V (G) \ Sm for m = 1, 2.
(`)
(`)
(`)
Proof. Let S` = S1 ∪ S2 where Sm ⊆ V (Gm ), m = 1, 2 and ` = 1, 2. In
(1)
(2)
view of Theorem 2 and Proposition 8 it suffices to show that |S1 | = |S1 | and
(1)
(2)
(1)
(2)
|S2 | = |S2 |. First, |S1 | ≥ 0 and |S1 | ≥ 0 (see Proposition 13). Since µ1 = λ1 is
(1)
(2)
a simple eigenvalue of G, according to (15) it follows that |S1 | = |S1 |. Therefore,
(1)
(1)
(2)
(2)
(1)
(2)
from |S1 | + |S2 | = |S1 | + |S2 | we obtain |S2 | = |S2 |.
u
t
Using Proposition 12 and following a similar procedure as in the proof of
Theorem 3, one may obtain the next statement.
Theorem 4. Let G be a connected graph with exactly two main eigenvalues and
let PGi (λ) = PGj (λ). Then P Gi (λ) = P Gj (λ).
(m)
Corollary 5. If σ(Gi ) = σ(Gj ) and σ(Gi ) = σ(Gj ), then xi
m = 1, 2, . . . , k.
(m)
= xj
for any
Proposition 18. Let G be a graph with n main eigenvalues µ1 , µ2 , . . . , µn and let
n
n
S
S
(1) (2) (1)
(2)
S1 =
Sm and S2 =
Sm . If |Sm | = ± |Sm | for m = 1, 2, . . . , n then
m=1
m=1
GS1 and GS2 are cospectral.
Proof. µ1 , µ2 , . . . , µn are simple main eigenvalues of G. Combining (15) with
(1)
(2)
the left-hand side of relation (20) we find that |Sm |2 = nm ρ2m (S1 ) and |Sm |2 =
2
nm ρm (S2 ). Consequently, we obtain that ρm (S1 ) = ρm (S2 ) for m = 1, 2, . . . , n,
which completes the proof.
u
t
Finally, in order to demonstrate some results from this paper we shall consider
the following graph: G = K2n+1 ∪(n+1)K1 , understanding that Kn and nK1 denote
the complete graph of order n and a graph with n isolated vertices, respectively.
One can see that K2n+1 ∪ (n + 1)K1 and K2n+1 ∪ (n + 1)K1 are integral5 for any
non-negative integer n. Namely, G and its complement G have two main eigenvalues
µ1 = 2n, µ2 = 0 and µ1 = 2n + 1, µ2 = − (n + 1), respectively. Moreover, we have
2
+12n+4
n2
that (i) n1 = 2n + 1 and n2 = n + 1; (ii) n1 = 8n 3n+2
and n2 = 3n+2
; (iii)
n+1
2n
2n
n
σ(G) = {2n, 0
, − 1 } and σ(G) = {2n + 1, 0 , − 1 , − (n + 1)}, where the
multiplicity of a multiple eigenvalue is given in the form of an exponent.
5 A graph is called integral if its spectrum consists entirely of integers. If G is integral, then
according to (3) its complement G is integral if and only if the main spectrum M(G) consists
only of integers.
84
Mirko Lepović
REFERENCES
1. D. Cvetković: Graphs and their spectra. Univ. Beograd, Publ. Elektrotehn. Fak.
Ser. Mat. Fiz., No. 356 (1971), 1–50.
2. D. Cvetković, M. Doob, H. Sachs: Spectra of graphs – Theory and applications.
3rd revised and enlarged edition, J. A. Barth Verlag, Heidelberg – Leipzig, 1995.
3. D. Cvetković, P. Rowlinson, S. Simić: Eigenspaces of graphs. Cambridge University Press, Cambridge, 1997.
4. M. Lepović: On formal products and spectra of graphs. Discrete Math., 188 (1998),
137–149.
5. M. Lepović: On formal products and the Seidel spectrum of graphs. Publ. Inst. Math.
(Belgrade), 63, (77) (1998), 37–46.
6. M. Lepović: Application of formal products on some compound graphs. Bull. Serb.
Acad. Sci. Arts (Sci. Math.), 23 (1998), 33–44.
7. M. Lepović: On formal products and generalized adjacency matrices. Bull. Serb.
Acad. Sci. Arts (Sci. Math.), 24 (1999), 51–66.
8. M. Lepović: On formal products and angle matrices of a graph. Discrete Math., 243
(2002), 151–160.
9. P. Rowlinson: The spectrum of a graph modified by the addition of a vertex. Univ.
Beograd. Publ. Elektrotehn. Fak. Ser. Mat., 3 (1992), 67–70.
Department of Mathematics,
Faculty of Science,
University of Kragujevac,
34000 Kragujevac,
Yugoslavia.
[email protected]
(Received January 23, 2002)