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MATCH Communications in Mathematical and in Computer Chemistry

MATCH Commun. Math. Comput. Chem. 60 (2008) 473-484

ISSN 0340 - 6253

DISTANCE EQUIENERGETIC GRAPHS H. S. Ramane,a I. Gutmanb and D. S. Revankara a

Department of Mathematics, Gogte Institute of Technology, Udyambag, Belgaum - 590008, India e-mail: [email protected], [email protected] b

Faculty of Science, University of Kragujevac, P. O. Box 60, 34000 Kragujevac, Serbia e-mail: [email protected] (Received December 2, 2007)

Abstract The distance energy ED (G) of a graph G is defined as the sum of the absolute values of the eigenvalues of the distance matrix of G. The graphs G1 and G2 are said to be distance equienergetic (D-equienergetic) if ED (G1 ) = ED (G2 ) . In this paper we obtain the eigenvalues of the distance matrix of the join of two graphs whose diameter is less than or equal to 2, and construct pairs of non D-cospectral, D-equienergetic graphs on n vertices for all n ≥ 9 .

- 474 INTRODUCTION In this paper we are concerned with simple graphs, that is graphs without loops, multiple edges or directed edges. Let G be such graph on n vertices and m edges. Let its vertices be labelled as v1 , v2 , . . . , vn . The distance between the vertices vi and vj , denoted by dij , is the length of the shortest path between them. The diameter of a graph G, denoted by diam(G) , is the maximum distance between any pair of vertices of G [4,12]. The distance matrix of a graph G is an n × n matrix D(G) = [dij ] . The characteristic polynomial of D(G) is defined as ψ(G : µ) = det(µI − D(G)) , where I is the identity matrix of order n . The eigenvalues of the distance matrix D(G), denoted by µ1 , µ2 , . . . , µn , are said to be the distance or D-eigenvalues of G and their collection is called the distance or D-spectrum of G . Two non-isomorphic graphs are said to be D-cospectral if they have same D-spectra [4,5,6]. Since the distance matrix is symmetric, its eigenvalues are real and can be ordered as µ1 ≥ µ2 ≥ · · · ≥ µn . The characteristic polynomial and the eigenvalues of the distance matrix of a graph were considered in [7–9,13,14,29]. The distance energy ED (G) of a graph G is defined as ED (G) =

n X

|µi | .

(1)

i=1

Eq. (1) was recently introduced by Indulal et al. [15] and was conceived in full analogy to the ordinary graph energy E(G) defined as [10,11] E(G) =

n X

|λi |

(2)

i=1

where λ1 , λ2 , . . . , λn are the eigenvalues of the adjacency matrix of G [6]. Several bounds for the distance energy of a graph are obtained in [15,20,21]. The graphs G1 and G2 are said to be equienergetic if E(G1 ) = E(G2 ) . Numerous results on (non-isomorphic) equienergetic graphs can be found in [1–3,16–19,23–28]. The connected graphs G1 and G2 are said to be D-equienergetic (distance equienergetic) if ED (G1 ) = ED (G2 ) . For obvious reason D-cospectral graphs are D- equienergetic. Therefore we are interested in non D-cospectral, D-equienergetic graphs having equal number of vertices. Indulal et al. [15] constructed pairs of D-equienergetic

- 475 graphs on n vertices for n ≡ 1 (mod 3) and for n ≡ 0 (mod 6) . Ramane, Revankar, Gutman and Walikar [22] proved that if G1 and G2 are r-regular graphs on n vertices and diam(Gi ) ≤ 2 , i = 1, 2 , then ED (Lk (G1 )) = ED (Lk (G2 )) for k ≥ 1, where Lk (G) is the k-th iterated line graph of G . In this paper we obtain the characteristic polynomial of the distance matrix of the join of two regular graphs whose diameter is less than or equal to 2 and thereby construct pairs of non D-cospectral, D-equienergetic graphs on n vertices for all n ≥ 9.

ON THE JOIN OF GRAPHS Definition [6, 12]. 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 ¡ @ @

G1

u ¡

@u

u @ @ @u ¡ ¡ ¡ u

u a !u ! ! ¡S a aa¶@ ! ¡!!S @ ¶ aa a !! S¶ a ¡ @u u a ! a @ a ¡ ¶S !!! a ! @ ¡ aa ¶ !S Su @u a ¡ ¶! a !

G2

G1 ∇G2 Fig. 1

Theorem 1. Let Gi be an ri -regular graph on ni vertices and diam(Gi ) ≤ 2, i = 1, 2 . Then the characteristic polynomial of the distance matrix of G1 ∇G2 is ψ(G1 ∇G2 : µ) =

[(µ − 2n1 + 2 + r1 )(µ − 2n2 + 2 + r2 ) − n1 n2 ] ψ(G1 : µ)ψ(G2 : µ) . (µ − 2n1 + 2 + r1 )(µ − 2n2 + 2 + r2 ) (3)

Proof. ψ(G1 ∇G2 : µ) = det(µI − D(G1 ∇G2 )) ¯ ¯ µI − D(G1 ) −Jn1 ×n2 = ¯¯ n1 −Jn2 ×n1 µIn2 − D(G2 ) where J is a matrix whose all entries are equal to unity.

¯ ¯ ¯ ¯

(4)

- 476 The determinant (4) can ¯ ¯ µ −d12 ¯ ¯ −d21 µ ¯ .. ¯ . ¯ ¯ ¯ −dn1 1 −dn1 2 ¯ −1 ¯ −1 ¯ −1 ¯ −1 ¯ .. ¯ . ¯ ¯ −1 −1

be written as . . . −d1n1 . . . −d2n1 .. .

−1 −1

−1 −1 −1 −d012 µ

... ... .. .

... ... ... .. .

µ −1 −1

−1 µ −d021

...

−1

−d0n2 1 −d0n2 2 . . .

−1 −1

. . . −1 . . . −d01n2 . . . −d02n2 .. . µ

¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯

(5)

where dij is the distance between the vertices vi and vj in G1 and d0ij is the distance between the vertices ui and uj in G2 . In Gi , every vertex is at distance one from ri vertices and at distance two from remaining ni − 1 − ri vertices. Therefore n1 X

dij = 2n1 − r1 − 2

for i = 1, 2, . . . , n1

(6)

d0ij = 2n2 − r2 − 2

for i = 1, 2, . . . , n2 .

(7)

j=1

and

n2 X j=1

We now perform the number of transformations that leave the value of the determinant (5) unchanged. Subtract the row (n1 + 1) from the rows (n1 + 2), (n1 + 3), . . . , (n1 + n2 ) of (5) to obtain (8): ¯ ¯ µ −d12 ¯ ¯ −d21 µ ¯ .. ¯ . ¯ ¯ ¯ −dn1 1 −dn1 2 ¯ −1 ¯ −1 ¯ 0 ¯ 0 ¯ .. ¯ . ¯ ¯ 0 0

. . . −d1n1 . . . −d2n1 .. . ... ... ... .. .

µ −1 0

...

0

−1 −1

−1 −1

−1 µ −d021 − µ

−1 −d012 µ + d012

... ... .. .

−1 −1

... −1 ... −d01n2 . . . −d02n2 + d01n2 .. .

−d0n2 1 − µ −d0n2 2 + d012 . . .

µ + d01n2

¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ . ¯ ¯ ¯ ¯ ¯ ¯ ¯

(8)

Adding the columns (n1 + 2), (n1 + 3), . . . , (n1 + n2 ) to the column (n1 + 1) of (8),

- 477 using Eq. (7), and noting that d0ij = d0ji we arrive at the determinant (9): ¯ ¯ µ −d12 . . . −d1n1 −n2 −1 ... −1 ¯ ¯ −d21 µ . . . −d2n1 −n2 −1 ... −1 ¯ .. .. .. ¯ . . . ¯ ¯ µ −n2 −1 ... −1 ¯ −dn1 1 −dn1 2 . . . ¯ −1 . . . −1 µ − 2n2 + 2 + r2 −d012 ... −d01n2 ¯ −1 ¯ 0 ... 0 0 µ + d012 . . . −d02n2 + d01n2 ¯ 0 ¯ .. .. .. ¯ . . . ¯ ¯ 0 0 0 ... µ + d0 0 ... 0 0 −d + d n2 2

which evidently ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ where

12

1n2

¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ (9) ¯ ¯ ¯ ¯ ¯ ¯ ¯

is equal to (10): µ −d21 .. .

−d12 µ

. . . −d1n1 . . . −d2n1 .. .

−dn1 1 −dn1 2 . . . −1 −1 . . .

µ −1

¯ ¯ ¯ ¯ ¯ ¯ ¯ |B| ¯ ¯ −n2 ¯ µ − 2n2 + 2 + r2 ¯ −n2 −n2

¯ ¯ µ + d0 −d023 + d013 . . . −d02n2 + d01n2 12 ¯ ¯ −d0 + d0 µ + d013 . . . −d03n2 + d01n2 32 12 ¯ |B| = ¯ .. .. ¯ . . ¯ 0 0 0 0 ¯ −dn 2 + d12 −dn 3 + d13 . . . µ + d01n2 2 2

¯ ¯ ¯ ¯ ¯ ¯ . ¯ ¯ ¯

(10)

(11)

In (10) the determinant is of order (n1 + 1) . Subtract the first row from the rows 2, 3, . . . , n1 , to obtain (12): ¯ ¯ µ −d12 ¯ ¯ −d21 − µ µ + d12 ¯ ¯ .. ¯ . ¯ ¯ −dn1 1 − µ −dn1 2 + d12 ¯ ¯ −1 −1

... −d1n1 . . . −d2n1 + d1n1 .. . ... ...

µ + d1n1 −1

¯ ¯ ¯ ¯ ¯ ¯ ¯ |B| . ¯ ¯ 0 ¯ µ − 2n2 + 2 + r2 . ¯ −n2 0

(12)

Adding columns 2, 3, . . . , n1 to the first column of (12) and using Eq. (6) we get (13): ¯ ¯ µ − 2n1 + 2 + r1 −d12 ¯ ¯ 0 µ + d12 ¯ ¯ .. ¯ . ¯ ¯ 0 −dn1 2 + d12 ¯ ¯ −n1 −1

... −d1n1 . . . −d2n1 + d1n1 .. . ... ...

µ + d1n1 −1

¯ ¯ ¯ ¯ ¯ ¯ ¯ |B| . (13) ¯ ¯ 0 ¯ µ − 2n2 + 2 + r2 ¯ −n2 0

Expand it along the first column to obtain (14): {(µ − 2n1 + 2 + r1 ) ∆1 − (−1)n1 n1 ∆2 } |B|

(14)

- 478 where ¯ ¯ µ + d12 −d23 + d13 ¯ ¯ −d32 + d12 µ + d13 ¯ ¯ .. ∆1 := ¯ . ¯ ¯ −dn1 2 + d12 −dn1 3 + d13 ¯ ¯ −1 −1 and

. . . −d2n1 + d1n1 . . . −d3n1 + d1n1 .. . ... ...

¯ ¯ −d12 −d13 ¯ ¯ µ + d12 −d23 + d13 ¯ ¯ µ + d13 ∆2 = ¯ −d32 + d12 ¯ . . ¯ . ¯ ¯ −dn1 2 + d12 −dn1 3 + d13

µ + d1n1 −1

¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ 0 ¯ µ − 2n2 + 2 + r2 ¯ 0 0

¯ ... −d1n1 −n2 ¯¯ . . . −d2n1 + d1n1 0 ¯¯ . . . −d3n1 + d1n1 0 ¯¯ . ¯ .. ¯ . ¯ ... µ + d1n1 0 ¯

The expression (14) can be rewritten as {(µ − 2n1 + 2 + r1 )(µ − 2n2 + 2 + r2 )|A| − n1 n2 |A|}|B| = |A||B|{(µ − 2n1 + 2 + r1 )(µ − 2n2 + 2 + r2 ) − n1 n2 }

(15)

where ¯ ¯ µ + d12 −d23 + d13 . . . −d2n1 + d1n1 ¯ ¯ −d32 + d12 µ + d13 . . . −d3n1 + d1n1 ¯ |A| = ¯ .. .. ¯ . . ¯ ¯ −dn1 2 + d12 −dn1 3 + d13 . . . µ + d1n1

¯ ¯ ¯ ¯ ¯ ¯ . ¯ ¯ ¯

(16)

The determinant (16) can be written as |A| =

1 × (µ − 2n1 + 2 + r1 )

¯ ¯ µ − 2n1 + 2 + r1 −d12 −d13 ¯ ¯ 0 µ + d12 −d23 + d13 ¯ ¯ 0 −d32 + d12 µ + d13 × ¯ ¯ .. ¯ . ¯ ¯ 0 −dn1 2 + d12 −dn1 3 + d13

... −d1n1 . . . −d2n1 + d1n1 . . . −d3n1 + d1n1 .. . ...

µ + d1n1

¯ ¯ ¯ ¯ ¯ ¯ ¯ . (17) ¯ ¯ ¯ ¯

From Eq. (6) the sum of the i-th row in (17) is µ + di1 for i = 2, 3, . . . , n1 . Therefore, by subtracting the columns 2, 3, . . . , n1 of (17) from the first column, we

- 479 obtain (18): |A| =

1 × (µ − 2n1 + 2 + r1 )

¯ ¯ µ −d12 −d13 ¯ ¯ −µ − d21 µ + d −d 12 23 + d13 ¯ ¯ µ + d13 × ¯ −µ − d31 −d32 + d12 ¯ .. ¯ . ¯ ¯ −µ − dn1 1 −dn1 2 + d12 −dn1 3 + d13

...

µ + d1n1

Add the first row of (18) to the rows 2, 3, . . . , n1 to obtain (19): ¯ ¯ µ −d12 −d13 . . . −d1n1 ¯ ¯ −d21 µ −d23 . . . −d2n1 ¯ 1 ¯ −d31 −d32 µ . . . −d3n1 |A| = ¯ (µ − 2n1 + 2 + r1 ) ¯ .. .. ¯ . . ¯ ¯ −dn1 1 −dn1 2 −dn1 3 . . . µ =

¯ ¯ ¯ ¯ ¯ ¯ ¯ . ¯ ¯ ¯ ¯

... −d1n1 . . . −d2n1 + d1n1 . . . −d3n1 + d1n1 .. .

1 ψ(G1 : µ) . (µ − 2n1 + 2 + r1 )

(18)

¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ (19)

In a similar manner we can show that from (11) follows |B| =

1 ψ(G2 : µ) . (µ − 2n2 + 2 + r2 )

Substituting (19) and (20) back into (15) yields Eq. (3).

(20) ¥

Theorem 2. Let Gi be an ri -regular graph on ni vertices and diam(Gi ) ≤ 2 , i = 1, 2 . Then ED (G1 ∇G2 ) = ED (G1 ) + ED (G2 ),

if XY ≥ n1 n2 p = ED (G1 ) + ED (G2 ) − (X + Y ) + (X + Y )2 − 4(XY − n1 n2 ), if XY < n1 n2

where X = 2n1 − 2 − r1 and Y = 2n2 − 2 − r2 . Proof. If Gi is an ri -regular graph on ni vertices and diam(Gi ) ≤ 2, i = 1, 2 , then from Theorem 1, ψ(G1 ∇G2 : µ) =

=

[(µ − 2n1 + 2 + r1 )(µ − 2n2 + 2 + r2 ) − n1 n2 ] ψ(G1 : µ)ψ(G2 : µ) (µ − 2n1 + 2 + r1 )(µ − 2n2 + 2 + r2 ) [(µ − X)(µ − Y ) − n1 n2 ] ψ(G1 : µ)ψ(G2 : µ) (µ − X)(µ − Y )

- 480 which gives that (µ − X)(µ − Y )ψ(G1 ∇G2 : µ) = (µ − X)(µ − Y )ψ(G1 : µ)ψ(G2 : µ) . Let P1 (µ) = (µ − X)(µ − Y ) ψ(G1 ∇G2 : µ) and P2 (µ) = (µ − X)(µ − Y ) ψ(G1 : µ) ψ(G2 : µ) . The roots of the equation P1 (µ) = 0 are X, Y and the D-eigenvalues of G1 ∇G2 . Therefore the sum of the absolute values of the roots of P1 (µ) = 0 is X + Y + ED (G1 ∇G2 ) .

(21)

The roots of P2 (µ) = 0 are the D-eigenvalues of G1 and G2 and ´ p 1 ³ X + Y ± (X + Y )2 − 4(XY − n1 n2 ) . 2 Therefore the sum of the absolute values of the roots of P2 (µ) = 0 is ¯ h i¯¯ p ¯1 ED (G1 ) + ED (G2 ) + ¯¯ X + Y + (X + Y )2 − 4(XY − n1 n2 ) ¯¯ 2 ¯ h i¯¯ p ¯1 + ¯¯ X + Y − (X + Y )2 − 4(XY − n1 n2 ) ¯¯ . 2

(22)

Since P1 (µ) = P2 (µ) , equating Eqs. (21) and (22) we get ED (G1 ∇G2 ) = ED (G1 ) + ED (G2 ) − (X + Y ) ¯ h i¯¯ p ¯1 + ¯¯ X + Y + (X + Y )2 − 4(XY − n1 n2 ) ¯¯ 2 ¯ h i¯¯ p ¯1 + ¯¯ X + Y − (X + Y )2 − 4(XY − n1 n2 ) ¯¯ . 2

(23)

Case 1: If XY ≥ n1 n2 , then Eq. (23) reduces to ED (G1 ∇G2 ) = ED (G1 ) + ED (G2 ) . Case 2: If XY < n1 n2 , then Eq. (23) reduces to ED (G1 ∇G2 ) = ED (G1 ) + ED (G2 ) − (X + Y ) +

p

(X + Y )2 − 4(XY − n1 n2 ) .

- 481 This completes the proof.

¥

Corollary 2.1. If H1 and H2 are non D-cospectral, D-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, ED (H1 ∇G) = ED (H2 ∇G).

¥

CONSTRUCTION OF DISTANCE EQUIENERGETIC GRAPHS Theorem 3. There exist pairs of non D-cospectral, D-equienergetic graphs on n vertices for all n ≥ 9 . Proof. Consider the graphs Ha and Hb as shown in Fig. 2.

u u ¢S L ¿¯A ¢ LS ¿¯ A A ¢ LS ¿ ¯ A ¢ L S¿ ¯ Au ¢ L ¯ S u ¿ à X X é é à ¢ AbX L ¯ X S ¿ à b à Xà à X © S u X © ¢ A b¿ L ¯ XBu L ¯ u ¢ A ¥ #c B A ¥ # cB ¢ Au cB¢ ¥# u

u ¢A @ ¡ ¡¢ A@ A @ ¡ ¢ u Au @ ¡©¢ ¡HHH @ ¡© @ © u © @u ¡ H ¡ @u H © ¡@ © @HH ¡ ¡ @u ©©¡ @ Hu ¢ ¡ @ A ¢ ¡ @ A @A ¢¡ Au ¢ @ ¡

Ha

Hb Fig. 2

By direct computation, ψ(Ha : µ) = (µ − 12)(µ + 3)4 µ4

(24)

ψ(Hb : µ) = (µ − 12)(µ + 4)(µ + 3)2 (µ + 1)2 µ3 .

(25)

and

Both Ha and Hb are regular graphs on 9 vertices and of degree 4. Also diam(Hi ) ≤ 2 , i = a, b , and ED (Ha ) = 24 = ED (Hb ) . Let H be any r-regular graph on p ≥ 1 vertices and diam(H) ≤ 2 . Then by Theorem 2 ED (Ha ∇H) = ED (Hb ∇H) = 24 + ED (H)

if 5p ≥ 8 + 4r

- 482 and ED (Ha ∇H) = ED (Hb ∇H) = ED (H) + 14 − 2p + r +

p

(10 + 2p − r)2 − 12(5p − 8 − 4r)

if 5p < 8 + 4r .

Thus from both cases, Ha ∇H and Hb ∇H are D-equienergetic. By Eqs. (24) and (25), Ha and Hb are non D-cospectral, so from Theorem 1, Ha ∇H and Hb ∇H are also non D-cospectral. Further Ha ∇H and Hb ∇H possesses equal number of vertices n = 9 + p , p = 1, 2, . . . . That the theorem holds also for n = 9 is directly verified from Eqs. (24) and (25). ¥ Let Kp be the complete graph on p vertices. It is regular of degree p − 1 and diam(Kp ) = 1. The adjacency matrix and the distance matrix of Kp are same. Therefore ED (Kp ) = E(Kp ) = 2(p − 1) [6,10]. Using this in Theorem 2 we have following result. Theorem 4. If Ha and Hb are the graphs as shown in Fig. 2, then ED (Ha ∇Kp ) = ED (Hb ∇Kp ) = 2(p + 11)

if p ≥ 4

and ED (Ha ∇Kp ) = ED (Hb ∇Kp ) = p + 11 +

p

(p + 11)2 − 12(p − 4)

if p < 4 .

¥

Conclusion. From Corollary 2.1 it is easy to construct a pair of non D-cospectral, D-equienergetic graphs. In particular from Theorem 3 and Theorem 4, it is easy to construct a pair of non D-cospectral, D-equienergetic n-vertex graphs for all n ≥ 9 .

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