Spec. Matrices 2017; 5:61–63
Research Article
Open Access
Wasin So
A shorter proof of the distance energy of
complete multipartite graphs
DOI 10.1515/spma-2017-0005
Received August 12, 2016; accepted October 11, 2016
Abstract: Caporossi, Chasser and Furtula in [Les Cahiers du GERAD (2009) G-2009-64] conjectured that the
distance energy of a complete multipartite graph of order n with r ≥ 2 parts, each of size at least 2, is equal
to 4(n − r). Stevanovic, Milosevic, Hic and Pokorny in [MATCH Commun. Math. Comput. Chem. 70 (2013), no.
1, 157-162.] proved the conjecture, and then Zhang in [Linear Algebra Appl. 450 (2014), 108-120.] gave another
proof. We give a shorter proof of this conjecture using the interlacing inequalities of a positve semi-definite
rank-1 perturbation to a real symmetric matrix.
Keywords: distance energy, multipartite graph, interlacing inequalities
MSC: 05C50, 15A18, 05C90
Given a connected graph G of order n ≥ 2 with vertex labels {1, 2, . . . , n}, define the distance matrix [3]
D(G) = [d ij ] of G by d ij = dist(i, j) ≥ 1 if i ≠ j, otherwise d ii = 0. Then D(G) is a real symmeric matrix with
zero diagonal entries. Hence the distance spectrum Sp D (G) of G is the collection of real eigenvalues of D(G),
called distance eigenvalues of G:
Sp D (G) = {λ1 ≥ λ2 ≥ · · · ≥ λ n } .
A comprehensive survey on distance spectra can be found in [1]. Next define the distance energy [5] E D (G) of
G as follows:
E D (G) = |λ1 | + |λ2 | + · · · + |λ n |.
Since trD(G) = 0, we have λ1 + · · · + λ n = 0, and so
E D (G) = 2
X
|λ i |.
λ i <0
This equivalent definition of distance energy is useful when positive eigenvalues are hard to compute. It is
interesting and also important to compute the distance energy for specific families of graphs. Caporossi, Chasser and Furtula in [2] conjectured the closed formula for the distance energy of complete multipartite graphs
(see Theorem 2). This conjectured formula was confirmed for the case of bipartite graphs by Stevanovic and
Indulal in [6]. Later Stevanovic, Milosevic, Hic and Pokorny in [7] proved the full conjecture. Recently, Zhang
in [8] gave another proof. However, both proofs are long and involved. In this note, we give a shorter proof
using the interlacing inequalities of a positve semi-definite rank-1 perturbation to a real symmetric matrix.
Lemma 1. Let A be an n × n real symmetric matrix. For any real vector x, we have the interlacing inequalities:
λ1 (A + xx T ) ≥ λ1 (A) ≥ λ2 (A + xx T ) ≥ · · · ≥ λ n−1 (A) ≥ λ n (A + xx T ) ≥ λ n (A),
where λ k (·) denotes the k-th eigenvalue of a real symmetric matrix when arranged from largest to smallest.
Wasin So: Department of Mathematics and Statistics, San Jose State University, San Jose, CA 95192, USA, E-mail:
[email protected]
© 2017 Wasin So, published by De Gruyter Open.
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.
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62 | Wasin So
Proof. Recall the Weyl’s inequalities for the sum of two real symmetric matrices A and B [4]:
λ i+j−1 (A + B) ≤ λ i (A) + λ j (B) ≤ λ i+j−n (A + B).
Take B = xx T a positive semi-definite rank-1 matrix with eigenvalues
λ1 (xx T ) = x T x,
λ2 (xx T ) = · · · = λ n (xx T ) = 0.
Then we have λ i (A + xx T ) ≥ λ i (A) for i = 1, . . . , n, and λ i (A) ≥ λ i+1 (A + xx T ) for i = 1, . . . , n − 1.
Theorem 2. Let K p1 ,p2 ,...,p r be a complete multipartite graph with 2 ≤ r and 2 ≤ p r ≤ · · · ≤ p1 . Then
E D (K p1 ,p2 ,...,p r ) = 4(p1 + p2 + · · · + p r − r).
Proof. First of all, note that the distance between non-adjacent vertices is always 2 in a complete multipartite
graph, and so the distance matrix of K p1 ,p2 ,...,p r is


2J p1 ,p1 − 2I p1
J p1 ,p2
···
J p1 ,p r


J p2 ,p1
2J p2 ,p2 − 2I p2 · · ·
J p2 ,p r



,
D(K p1 ,p2 ,...,p r ) = 
..
..
..
..

.


.
.
.
J p r ,p1
J p r ,p2
· · · 2J p r ,p r − 2I p r
where J a,b is the a×b matrix with all entries equal to 1, and I k is the k×k identity matrix. Let n = p1 +p2 +· · ·+p r ,
then we have


2J p1 ,p1 J p1 ,p2 · · · J p1 ,p r
 J

 p2 ,p1 2J p2 ,p2 · · · J p2 ,p r 
,
D(K p1 ,p2 ,...,p r ) + 2I n = 
..
..
..
..


.


.
.
.
J p r ,p1
J p r ,p2 · · · 2J p r ,p r
which is easily seen to have r different and linearly independent rows. Hence
rank(D(K p1 ,p2 ,...,p r ) + 2I n ) = r,
and so D(K p1 ,p2 ,...,p r ) + 2I n has exactly r nonzero eigenvalues. On the other hand,


J p1 ,p1
0
···
0
 0
J p2 ,p2 · · ·
0 


D(K p1 ,p2 ,...,p r ) + 2I n = 
..
..
.. 
..

 + Jn ,
.

.
.
. 
0
0
· · · J p r ,p r


J p1 ,p1
0
···
0
 0
J p2 ,p2 · · ·
0 


which is a rank-1 perturbation. Note that the spectrum of 
..
..
.. 
..

 is the disjoint union
.

.
.
. 
0
0
· · · J p r ,p r
n
o
n
o
(p i −1)
(n−r)
of the spectrum of J p i ,p i = p i , 0
, i.e., p1 , p2 , · · · , p r , 0
. Let the eigenvalues of D(K p1 ,p2 ,...,p r ) +
2I n be µ1 ≥ µ2 ≥ · · · ≥ µ n . By Lemma 1, we have
µ1 ≥ p1 ≥ µ2 ≥ p2 ≥ · · · µ r ≥ p r ≥ µ r+1 ≥ 0 ≥ µ r+2 ≥ 0 · · · 0 ≥ µ n ≥ 0.
Consequently, µ1 , µ2 , · · · , µ r ≥ 2, p r ≥ µ r+1 ≥ 0, and µ r+2 = · · · = µ n = 0. However, rank(D(K p1 ,p2 ,...,p r ) +
2I n ) = r, we must have µ r+1 = 0 too. Therefore, the spectrum of D(K p1 ,p2 ,...,p r ) is
o
n
µ1 − 2, µ2 − 2, · · · , µ r − 2, −2(n−r) ,
where µ i − 2’s are non-negative eigenvalues, and −2 is the only negative eigenvalue of multiplicity n − r.
Finally,
E D (K p1 ,p2 ,...,p r ) = 2(| − 2| + · · · (n − r times) · · · + | − 2|) = 4(n − r).
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A shorter proof of the distance energy of complete multipartite graphs |
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References
[1] M. Aouchiche and P. Hansen, Distance spectra of graphs: a survey, Linear Algebra Appl. 458 (2014), 301–386.
[2] G. Caporossi, E. Chasset, B. Furtula, Some conjectures and properties on distance energy, Les Cahiers GERAD (2009) G-200964.
[3] R. Graham and H. Pollak, On the addressing problem for loop switching, Bell System Tech. J. 50 (1971), 2495–2519.
[4] R. Horn and C. Johnson, Matrix Analysis, Cambridge Univeristy Press, Cambridge, 1985.
[5] G. Indulal, I. Gutman and A. Vijayakumar, On distance energy of graphs, MATCH Commun. Math. Comput. Chem. 60 (2008),
no. 2, 461–472.
[6] D. Stevanovic and G. Indulal, The distance spectrum and energy of the compositions of regular graphs, Appl. Math. Lett. 22
(2009), no. 7, 1136–1140.
[7] D. Stevanovic, M. Milosevic, P. Hic and M. Pokorny, Proof of a conjecture on distance energy of complete multipartite graphs,
MATCH Commun. Math. Comput. Chem. 70 (2013), no. 1, 157-162
[8] X. Zhang, The inertia and energy of distance matrices of complete k-partite graphs, Linear Algebra Appl. 450 (2014), 108-120.
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