Complementary Distance Equienergetic Graphs Harishchandra S. Ramane and Mahadevappa M. Gundloor Department of Mathematics, Karnatak University, Dharwad - 580003, India E-mails:
[email protected],
[email protected]
Abstract The complementary distance matrix of a graph G is defined as CD(G) = [cij ], in which cij = 1+D−dij if i 6= j and cij = 0 if i = j, where D is the diameter of G and dij is the distance between the vertices vi and vj in G. The complementary distance energy CDE(G) of G is defined as the sum of the absolute values of the eigenvalues of the complementary distance matrix of G. Two graphs G1 and G2 are said to be CD-equienergetic if CDE(G1 ) = CDE(G2 ). In this paper we obtain the CD-eigenvalues and CDenergy of the join of some regular graphs and thus construct the non CD-cospectral, CD-equienergetic graphs on n vertices, for all n ≥ 6. Mathematics Subject Classification: 05C50, 05C12 Keywords: Complementary distance eigenvalues, complementary distance energy, join of graphs.
1
Introduction:
Let G be a simple, connected graph with n vertices and m edges. Let the vertex set of G be V (G) = {v1 , v2 , . . . , vn }. The distance between the vertices vi and vj , denoted by dij = d(vi , vj ) is the length of shortest path joining them. The diameter of a graph G, denoted by diam(G), is the maximum distance between any pair of vertices of G [1]. The complementary distance between the vertices vi and vj is defined as cij = 1 + D − dij , where D is the diameter of G and dij is the distance between the vertices vi and vj in G. The complementary distance (CD) matrix [3] of a graph G is an n × n real symmetric matrix CD(G) = [cij ], where ( 1 + D − dij , if i 6= j cij = 0, otherwise. The complementary distance matrix gives quantitative structure property relationship (QSPR) models in chemistry [3, 4]. The characteristic polynomial of CD(G) is defined as ψ(G : µ) = det(µI − CD(G)), where I is the identity matrix of order n. The eigenvalues of the complementary distance matrix, denoted by µ1 , µ2 , . . . , µn are said to be the complementary distance eigenvalues or CD-eigenvalues of G and their collection is called the CD-spectra of G. Two non-isomorphic graphs are said to be CD-cospectral if they have same CD-spectra. The complementary distance energy or CD-energy of a graph G denoted by CDE(G) is defined as CDE(G) =
n X i=1
1
|µi | .
(1)
The Eq. (1) is in full analogy to the ordinary graph energy [2] defined as the sum of the absolute values of the eigenvalues of the adjacency matrix of G. For more details about the graph energy we refer [5]. The graphs G1 and G2 are said to be complementary distance equienergetic or CD-equienergetic if CDE(G1 ) = CDE(G2 ). Obviously the CD-cospectral graphs are CD-equienergetic. Therefore it is interesting to find non CD-cospectral, CD-equienergetic graphs having equal number of vertices. In [6] the CD-energy of line graphs of certain regular graphs was obtained and thus the pairs of CD-equienergetic graphs were obtained. Here we obtain the characteristic polynomial of the CD-matrix of the join of two regular graphs whose diameter is less than or equal to two and thereby construct pairs of non CD-cospectral, CD-equienergetic graphs on n vertices for all n ≥ 6.
2
CD-spectra and CD-energy of join of some graphs:
Definition: The join of two graphs G1 and G2 , denoted by G1 ∇G2 , is a graph obtained from G1 and G2 by joining each vertex of G1 to all vertices of G2 . u
u @
u u H @ A HH A H@ HH A @u A A Au u
@ @u u G1
u
G1 ∇G2
G2 Fig. 1
Theorem 2.1. Let Gi be an ri -regular graph on ni vertices and diam(Gi ) = 2, i = 1, 2. Then the characteristic polynomial of the complementary distance matrix of G1 ∇G2 is ψ(G1 ∇G2 : µ) =
[(µ − X)(µ − Y ) − 4n1 n2 ] ψ(G1 : µ)ψ(G2 : µ), (µ − X)(µ − Y )
(2)
where X = n1 + r1 − 1 and Y = n2 + r2 − 1. Proof. ψ(G1 ∇G2 : µ)
=
det(µI − CD(G1 ∇G2 ))
µIn1 − CD(G1 ) (−2)Jn1 ×n2 = (−2)Jn2 ×n1 µIn2 − CD(G2 ) where J is the matrix whose all entries are equal to unity.
2
(3)
The determinant (3) can be written as µ −c12 · · · −c21 µ ··· .. .. . . −cn1 1 −cn1 2 · · · −2 −2 ··· −2 −2 ··· . .. . . . −2 −2 ···
−c1n1 −c2n1
−2 −2
−2 −2
··· ··· .. .
−2 −2
µ −2 −2
−2 µ −c021
−2 −c012 µ
··· ··· ··· .. .
−2 −c01n2 −c02n2
−2
−c0n2 1
−c0n2 2
···
µ
(4)
where cij is the complementary distance between the vertices vi and vj in G1 and c0ij is the complementary distance between the vertices ui and uj in G2 . Since Gi is an ri -regular graph and diam(Gi ) = 2, every vertex of Gi is at distance one from ri vertices and at distance 2 from remaining (ni − 1 − ri ) vertices, i = 1, 2. Therefore n1 X
cij = n1 + r1 − 1
for i = 1, 2, . . . , n1
(5)
c0ij = n2 + r2 − 1
for i = 1, 2, . . . , n2 .
(6)
j=1
and
n2 X j=1
Performing following operations in the sequence on the determinant (4) we get (7). (i) Subtract the row (n1 + 1) from the rows (n1 + 2), (n1 + 3), . . . , (n1 + n2 ). (ii) Add the columns (n1 + 2), (n1 + 3), . . . , (n1 + n2 ) to the column (n1 + 1) and apply Eq. (6), where Y = n2 + r2 − 1. µ −c21 .. . −cn 1 1 −2
−c12 µ −cn1 2 −2
··· ··· .. .
−c1n1 −c2n1
··· ···
µ −2
|B| −2n2 µ−Y −2n2 −2n2
(7)
where µ + c012 −c032 + c012 |B| = .. . −c0 + c0 n2 2 12
−c023 + c013 µ + c013
··· ··· .. .
−c02n2 + c01n2 −c03n2 + c01n2
−c0n2 3 + c013
···
µ + c01n2
.
The first determinant in (7) is of order (n1 + 1) . Performing following operations sequentially on the first determinat of (7) we get (9). (i) Subtract the first row from the rows 2, 3, . . . , n1 . (ii) Add columns 2, 3, . . . , n1 to the first column and apply Eq. (5), where X = n1 + r1 − 1
3
(8)
µ−X 0 .. . 0 −2n1
−c12 µ + c12 −cn1 2 + c12 −2
··· ··· .. .
−c1n1 −c2n1 + c1n1
··· ···
µ + c1n1 −2
|B| . 0 µ−Y −2n2 0
(9)
Expand it along the first column to obtain (10): {(µ − X) ∆1 − (−1)n1 (2n1 )∆2 } |B|
(10)
where µ + c12 −c32 + c12 .. ∆1 = . −cn 2 + c12 1 −2 and
−c12 µ + c12 ∆2 = −c32 + c12 .. . −cn 2 + c12 1
0 µ−Y
−c23 + c13 µ + c13
··· ··· .. .
−c2n1 + c1n1 −c3n1 + c1n1
−cn1 3 + c13 −2
··· ···
µ + c1n1 −2
−c13 −c23 + c13 µ + c13
··· ··· ··· .. .
−c1n1 −c2n1 + c1n1 −c3n1 + c1n1
−2n2 0 0
−cn1 3 + c13
···
µ + c1n1
0
0 0
.
The expression (10) can be written as
(µ − X)(µ − Y ) |A| − (−1)n1 (2n1 )(−1)1+n1 (−2n2 )|A| |B|
= {(µ − X)(µ − Y ) − 4n1 n2 } |A||B| where
µ + c12 −c32 + c12 |A| = .. . −cn 2 + c12 1
(11)
−c23 + c13 µ + c13
··· ··· .. .
−c2n1 + c1n1 −c3n1 + c1n1
−cn1 3 + c13
···
µ + c1n1
.
(12)
The determinant (12) can be written as µ−X 0 1 0 |A| = (µ − X) .. . 0
−c12 µ + c12 −c32 + c12
−c13 −c23 + c13 µ + c13
··· ··· ··· .. .
−c1n1 −c2n1 + c1n1 −c3n1 + c1n1
−cn1 2 + c12
−cn1 3 + c13
···
µ + c1n1
.
(13)
By Eq. (5), the sum of the i-th row in (13) is µ + ci1 for i = 2, 3, . . . , n1 . Therefore, by subtracting the columns 2, 3, . . . , n1 of (13) from the first column, we obtain (14):
4
µ −µ − c21 1 −µ − c31 |A| = (µ − X) .. . −µ − cn 1 1
−c12 µ + c12 −c32 + c12
−c13 −c23 + c13 µ + c13
··· ··· ··· .. .
−c1n1 −c2n1 + c1n1 −c3n1 + c1n1
−cn1 2 + c12
−cn1 3 + c13
···
µ + c1n1
.
(14)
Add the first row to the rows 2, 3, . . . , n1 in (14) to obtain (15):
|A| =
=
µ −c21 1 −c31 (µ − X) .. . −cn 1 1
−c12 µ −c32
−c13 −c23 µ
··· ··· ··· .. .
−c1n1 −c2n1 −c3n1
−cn1 2
−cn1 3
···
µ
1 ψ(G1 : µ) . (µ − X)
(15)
In a similar manner we can show that from (8), |B| =
1 ψ(G2 : µ) . (µ − Y )
(16)
Substituting (15) and (16) back into (11) gives Eq. (2). Theorem 2.2. Let Kp be the complete graph on p vertices. Let G be an r-regular graph on n vertices and diam(G) = 2. Then the characteristic polynomial of the complementary distance matrix of Kp ∇G is ψ(Kp ∇G : µ) =
[(µ − 2p + 2)(µ − Z) − 4pn] (µ + 2)p−1 ψ(G : µ), (µ − Z)
(17)
where Z = n + r − 1. Proof. ψ(Kp ∇G : µ)
=
det(µI − CD(Kp ∇G))
(µ + 2)Ip − (2)Jp×p = (−2)Jn×p
(−2)Jp×n µIn − CD(G)
(18)
where J is the matrix whose all entries are equal to unity. The determinant (18) can be written as
µ −2 .. .
−2 µ
··· ··· .. .
−2 −2
−2 −2
−2 −2
··· ··· .. .
−2 −2 −2 .. .
−2 −2 −2
··· ··· ··· .. .
µ −2 −2
−2 µ −c21
−2 −c12 µ
··· ··· ··· .. .
−2
−2
···
−2
−cn1
−cn2
···
5
−2 −c1n −c2n µ −2 −2
(19)
where cij is the complementary distance between the vertices vi and vj in G. Since G is an r-regular graph and diam(G) = 2, every vertex of G is at distance one from r vertices and at distance 2 from remaining (n − 1 − r) vertices. Therefore n X
cij = n + r − 1
for i = 1, 2, . . . , n.
(20)
j=1
Performing row and column operations on the determinant (19) we get the (17). Theorem 2.3. Let Gi be an ri -regular graph on ni vertices and diam(Gi ) = 2, i = 1, 2. Then p CDE(G1 ∇G2 ) = CDE(G1 ) + CDE(G2 ) − (X + Y ) + (X − Y )2 + 16n1 n2 , where X = n1 + r1 − 1 and Y = n2 + r2 − 1. Proof. From Theorem 2.1, ψ(G1 ∇G2 : µ) =
[(µ − X)(µ − Y ) − 4n1 n2 ] ψ(G1 : µ)ψ(G2 : µ) (µ − X)(µ − Y )
which gives that (µ − X)(µ − Y )ψ(G1 ∇G2 : µ) = [(µ − X)(µ − Y ) − 4n1 n2 ] ψ(G1 : µ)ψ(G2 : µ). Let P1 (µ) = (µ − X)(µ − Y ) ψ(G1 ∇G2 : µ) and P2 (µ) = [(µ − X)(µ − Y ) − 4n1 n2 ] ψ(G1 : µ)ψ(G2 : µ). The roots of P1 (µ) = 0 are X, Y and the CD-eigenvalues of G1 ∇G2 . Therefore the sum of the absolute values of the roots of P1 (µ) = 0 is X + Y + CDE(G1 ∇G2 ).
(21)
The roots of P2 (µ) = 0 are CD-eigenvalues of G1 and G2 and i p 1h (X + Y ) ± (X + Y )2 − 4(XY − 4n1 n2 ) . 2 Therefore the sum of the absolute values of the roots of P2 (µ) = 0 is
CDE(G1 )
+
h i p 1 2 CDE(G2 ) + (X + Y ) + (X + Y ) − 4(XY − 4n1 n2 ) 2
+
h i p 1 (X + Y ) − (X + Y )2 − 4(XY − 4n1 n2 ) . 2
(22)
Since P1 (µ) = P2 (µ), equating (21) and (22), we get CDE(G1 ∇G2 )
= CDE(G1 ) + CDE(G2 ) − (X + Y ) h i p 1 2 + (X + Y ) + (X + Y ) − 4(XY − 4n1 n2 ) 2 h i p 1 2 + (X + Y ) − (X + Y ) − 4(XY − 4n1 n2 ) . 2 6
(23)
Since r1 ≤ n1 − 1 and r2 ≤ n2 − 1, XY = (n1 + r1 − 1)(n2 + r2 − 1) ≤ (2n1 − 2)(2n2 − 2) < 4n1 n2 . Therefore Eq. (23) reduces to CDE(G1 ∇G2 )
= CDE(G1 ) + CDE(G2 ) − (X + Y ) +
p (X + Y )2 − 4(XY − 4n1 n2 )
= CDE(G1 ) + CDE(G2 ) − (X + Y ) +
p (X − Y )2 + 16n1 n2 .
Corollary 2.4. If H1 and H2 are non CD-cospectral, CD-equienergetic regular graphs on n vertices and of same degree and diam(Hi ) = 2, i = 1, 2, then for any regular graph G with diam(G) = 2, CDE(H1 ∇G) = CDE(H2 ∇G). By Eq. (17), following result is obtained. Theorem 2.5. Let Kp be the complete graph on p vertices. Let G be an r-regular graph on n vertices and diam(G) = 2. Then p CDE(Kp ∇G) = CDE(G) + 2p − n − r − 1 + (2p − n − r − 1)2 + 16pn.
3
Construction of CD-equienergetic graphs:
Theorem 3.1. There exist a pair of non CD-cospectral, CD-equienergetic graphs on n vertices for all n ≥ 6. Proof. Consider the graphs Ha and Hb as shown in the Fig. 2. u H HH Hu u aa ! aa!!! !!aaa u au !! HH H Hu
u H HH Hu u
Ha
Hb
u HH
u H Hu
Fig. 2 By direct computation, ψ(Ha : µ) = (µ − 8)(µ + 4)(µ + 1)4
(24)
ψ(Hb : µ) = µ(µ − 8)(µ + 1)(µ + 3)2 .
(25)
and
Both Ha and Hb are regular graphs on 6 vertices and of degree 3. Also diam(Hi ) = 2, i = a, b, and CDE(Ha ) = CDE(Hb ) = 16. Then by Theorem 2.5,
7
CDE(Kp ∇Ha ) = CDE(Kp ∇Hb ) = 6 + 2p +
p
(2p − 10)2 + 96p.
Thus, Kp ∇Ha and Kp ∇Hb are CD-equienergetic graphs. By Eqs. (24) and (25), Ha and Hb are non CD-cospectral, so from Theorem 2.2, Kp ∇Ha and Kp ∇Hb are also non CD-cospectral. Further Kp ∇Ha and Kp ∇Hb possesses equal number of vertices n = 6 + p, p = 1, 2, ..... That the theorem holds also for n = 6 is directly verified from Eqs. (24) and (25). Using Theorem 2.3 we have folowing. Theorem 3.2. Let H be any r-regular graph on p ≥ 1 vertices and diam(H) = 2. If Ha and Hb are the graphs as shown in Fig. 2, then p CDE(Ha ∇H) = CDE(Hb ∇H) = CDE(H) + 9 − p − r + (9 − p − r)2 + 96p.
4
Conclusion:
From Corollary 2.4 and Theorem 3.2, it is easy to construct a pair of non CD-cospectral, CD-equienergetic graphs. In particular from Theorem 3.1, it is easy to construct a pair of non CD-cospectral, CD-equienergetic n-vertex graphs for all n ≥ 6. Acknowledgement: M. M. Gundloor is thankful to the Karnatak University, Dharwad for support through UGC-UPE scholarship No. KU/Sch/UGC-UPE/2013-14/1136.
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