Let K n1,n2,...,np denote the complete p-partite graph, p &gt; 1, on n = n 1 + n 2 + · · · + n p vertices and let n 1 ≥ n 2 ≥ · · · ≥ n p &gt; 0. We show that for a fixed value of n, both the spectral radius and the energy of complete p-partite graphs are minimal for complete split graph CS(n, p − 1) and are maximal for Turán graph T (n, p)

Gutman, Ivan

Rehman, Masood U.

Stevanović, Dragan

Ars Mathematica Contemporanea

Let Kn1;n2...;np denote the complete p-partite graph, p > 1, on n = n1+n2+...+np vertices and let n1≥ n2 ≥ ...≥ np > 0. We show that for a fixed value of n, both the spectral radius and the energy of complete p-partite graphs are minimal for complete split graph CS(n; p - 1) and are maximal for Tuŕan graph T(n; p)

Stevanovic, Dejan

Rehman M.

SCIDAR - Дигитална архива Универзитета у Крагујевцу

On spectral radius and energy of complete multipartite graphs

Repository of University of Primorska

                                                     Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 9 (2015) 109–113  On spectral radius and energy of complete multipartite graphs  Dragan Stevanović,  Ivan Gutman, Masood U. Rehman  Abstract  Let Kn1,n2,…,np denote the complete p-partite graph, p > 1, on n = n1+n2+ … +np vertices and let n1 ≥ n2 ≥ … ≥ np > 0. We show that for a fixed value of n, both the spectral radius and the energy of complete p-partite graphs are minimal for complete split graph CS(n, p –1) and are maximal for Turán graph T(n, p).  Keywords: Spectral radius of graph, graph energy, complete multipartite graph, complete split graph, Turán graph.  Math. Subj. Class.: 05C50           O spektralnem radiju in energiji polnih večdelnih grafov   Povzetek  Naj Kn1,n2,…,np označuje polni p-delni graf, p > 1, na n = n1+n2+ … +np vozliščih in naj bo n1 ≥ n2 ≥ … ≥ np > 0. Pokažemo, da sta pri fiksni vrednosti n tako spektralni radij kot tudi energija polnega p-delnega grafa minimalna za polni razcepljeni graf CS(n, p –1) in da sta maksimalna za Turánov graf T(n, p).  Ključne besede: Spektralni radij grafa, energija grafa, polni večdelni graf, polni razcepljeni graf, Turánov graf.  

Dragan Stevanović

Ivan Gutman

Masood U Rehman

CiteSeerX

Also available at http://amc-journal.eu
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 9 (2015) 109–113
On spectral radius and energy of complete
multipartite graphs
Dragan Stevanović ∗
University of Primorska, UP IAM, Muzejski trg 2, SI-6000 Koper, Slovenia and
Mathematical Institute, Serbian Academy of Science and Arts, Knez Mihajlova 36, 11000
Belgrade, Serbia
Ivan Gutman
Faculty of Science, University of Kragujevac, P.O. Box 60, 34000 Kragujevac, Serbia
Masood U. Rehman
Air University, Islamabad, Pakistan
Received 18 June 2013, accepted 8 March 2014, published online 8 August 2014
Abstract
LetKn1,n2,...,np denote the complete p-partite graph, p > 1, on n = n1+n2+ · · ·+np
vertices and let n1 ≥ n2 ≥ · · · ≥ np > 0. We show that for a fixed value of n, both the
spectral radius and the energy of complete p-partite graphs are minimal for complete split
graph CS(n, p− 1) and are maximal for Turán graph T (n, p).
Keywords: Spectral radius of graph, graph energy, complete multipartite graph, complete split graph,
Turán graph.
Math. Subj. Class.: 05C50
1 Introduction
Let G be a simple graph on n vertices, and λ1 ≥ λ2 ≥ · · · ≥ λn be its eigenvalues (i.e.,
the eigenvalues of the (0,1)-adjacency matrix of G) [2, 4]. Then λ1 = λ1(G) is said to be
the spectral radius of the graph G whereas
E = E(G) =
n∑
i=1
|λi| (1.1)
∗Corresponding author. The work of DS was supported by Research Programme P1-0285 and Research Project
J1-4021 of the Slovenian Research Agency and Research Grant ON174033 of the Ministry of Education and
Science of Serbia.
E-mail addresses: dragan.stevanovic@upr.si (Dragan Stevanović), gutman@kg.ac.rs (Ivan Gutman),
(Masood U. Rehman)
cb This work is licensed under http://creativecommons.org/licenses/by/3.0/
110 Ars Math. Contemp. 9 (2015) 109–113
is its energy. The spectral radius is one of the most thoroughly investigated graph-spectral
parameters [2, 4, 3]. Also the graph energy has recently attracted much attention [6, 7]. In
spite of this, not much is known on λ1 and E of complete multipartite graphs [5].
Let p > 1. Denote byKn1,n2,...,np the complete p-partite graph on n = n1+n2+ · · ·+
np vertices. For convenience, we assume that n1 ≥ n2 ≥ · · · ≥ np > 0 . Two particular
types of complete multipartite graphs are:
• Complete split graph CS(n, p − 1) = Kn−p+1,1,...,1 consisting of an independent
set of n − p + 1 vertices and a clique of p − 1 vertices, such that each vertex of the
independent set is adjacent to each vertex of the clique, and
• Turán graph T (n, p) ∼= Kdn/pe,...,dn/pe,bn/pc,...,bn/pc, the (p + 1)-clique-free graph
with maximum number of edges [10].
A fundamental result on the spectrum of complete multipartite graphs is:
Theorem 1.1 ([9, 8]). A connected graph has exactly one positive eigenvalue of its adja-
cency matrix if and only if it is a complete multipartite graph.
An immediate consequence of Theorem 1.1 is
Lemma 1.2. If λ1 is the spectral radius of the complete multipartite graph Kn1,n2,...,np ,
then E(Kn1,n2,...,np) = 2λ1 .
Proof. Since all graph eigenvalues are real numbers and their sum is zero, from Eq. (1.1)
follows that E(G) is equal to twice the sum of positive eigenvalues.
It is known that the characteristic polynomial of Kn1,n2,...,np is given by [2, 5]
φ(Kn1,n2,...,np , λ) = λ
n−p
(
1−
p∑
i=1
ni
λ+ ni
)
p∏
j=1
(λ+ nj) . (1.2)
The spectrum of Kn1,n2,...,np consists of the spectral radius λ1 determined from the equa-
tion
∑p
i=1
ni
λ+ni
= 1, eigenvalue 0 with multiplicity n−p and p−1 eigenvalues situated in
the intervals [−np,−np−1], . . . , [−n2,−n1]. In the special case n1 = n2 = · · · = np = t,
the spectrum of Kt,t,...,t consists of the spectral radius t(p − 1) with unit multiplicity,
eigenvalue 0 with multiplicity p(t− 1), and eigenvalue −t with multiplicity p− 1, so that
E(Kt,t,...,t) = 2(p− 1)t .
Remark 1.3. The 1-partite complete graph (when p = 1 and t = n) is the edgeless graph
Kn for which, consistently, λ1 = E = 0. The n-partite complete graph (when p = n
and t = 1) is the ordinary complete graph Kn for which, consistently, λ1 = n − 1 and
E = 2(n− 1).
If p = 2, then the special case of Eq. (1.2) is
φ(Kn1,n2 , λ) = λ
n−2
(
λ2 − n1 n2
)
,
from which the (well known) expressions for spectral radius and energy follow:
λ1(Kn1,n2) =
√
n1 n2, E(Kn1,n2) = 2
√
n1 n2 .
Because n1 + n2 = n , for a fixed number of vertices n, we arrive at:
D. Stevanović et al.: On spectral radius and energy of complete multipartite graphs 111
Claim 1.4. 1◦ λ1(Kn1,n2) and E(Kn1,n2) are minimal if n1 = n− 1 and n2 = 1.
2◦ λ1(Kn1,n2) and E(Kn1,n2) are maximal if n1 − n2 ≤ 1.
If p = 3, then the special case of Eq. (1.2) is
φ(Kt1,t2,t3 , λ) = λ
p−3
(
λ3 − (t1 t2 + t2 t3 + t3 t1)λ− 2 t1 t2 t3
)
. (1.3)
Using (1.3), it is relatively easy to show the following:
Claim 1.5. 1◦ Let n1+n2+n3 be equal to a fixed integer n. Then the spectral radius and
the energy of Kn1,n2,n3 are minimal if n1 = n− 2 and n2 = n3 = 1.
2◦ Let n1 + n2 + n3 be equal to a fixed integer n. Then the spectral radius and the energy
of Kn1,n2,n3 are maximal if n1 − n3 ≤ 1.
The above claims were the motivation for establishing our main results:
Theorem 1.6. Let p ≥ 2 and n1 + n2 + · · · + np be equal to a fixed integer n. Then the
spectral radius and the energy ofKn1,n2,...,np are minimal ifKn1,n2,...,np ∼= CS(n, p−1).
Theorem 1.7. Let p ≥ 2 and n1 + n2 + · · · + np be equal to a fixed integer n. Then the
spectral radius and the energy of Kn1,n2,...,np are maximal if Kn1,n2,...,np ∼= T (n, p).
2 Proofs of Theorems 1.6 and 1.7
Let λ1 and x be, respectively, the spectral radius and the corresponding unit eigenvector of
the adjacency matrix of Kn1,...,np . Since λ1 is a simple eigenvalue of Kn1,...,np , similar
vertices have equal x-components. Hence, we may denote by xi the common x-component
of vertices in the part ofKn1,...,np having ni vertices for i = 1, . . . , p. From the eigenvalue
equation, we have:
λ1 xi =
p∑
k=1
k 6=i
nk xk = X − ni xi
where X =
∑p
k=1 nk xk . Then
xi =
X
λ1 + ni
. (2.1)
Lemma 2.1. If ni − nj ≥ 2, then
λ1(Kn1,...,ni−1,...,nj+1,...,np) > λ1(Kn1,...,ni,...,nj ,...,np) .
Proof. Let λ1, x and E denote, respectively, the spectral radius, the corresponding eigen-
vector, and the edge set ofKn1,...,ni,...,nj ,...,np , and let λ
∗
1,A
∗ andE∗ denote, respectively,
the spectral radius, the adjacency matrix, and the edge set of Kn1,...,ni−1,...,nj+1,...,np .
From the Variational theorem we have
λ∗1 ≥ xTA∗x =
∑
uv∈E∗
2xuxv
=
∑
uv∈E
2xuxv +
∑
uv∈E∗\E
2xuxv −
∑
uv∈E\E∗
2xuxv
= λ1 + 2xi(ni − 1)xi − 2xinjxj
= λ1 + 2xiX
(
ni − 1
λ1 + ni
− nj
λ1 + nj
)
(2.2)
112 Ars Math. Contemp. 9 (2015) 109–113
by Eq. (2.1). Next, note that Kn1,...,ni,...,nj ,...,np has Kni,nj as an induced subgraph, so
that, by the Interlacing theorem [2],
λ1 ≥
√
ni nj > nj .
Therefore,
ni − 1
λ1 + ni
− nj
λ1 + nj
=
(ni − nj − 1)λ1 − nj
(λ1 + ni)(λ1 + nj)
≥ λ1 − nj
(λ1 + ni)(λ1 + nj)
> 0
so that λ∗1 > λ1 follows from Eq. (2.2).
Proof of Theorem 1.6 Let Km1,...,mp be the complete multipartite graph with the small-
est spectral radius. If there are two parametersmi ≥ mj ≥ 2, then (mi+1)−(mj−1) ≥ 2
and from Lemma 2.1
λ1(Km1,...,mi,...,mj ,...,mp) > λ1(Km1,...,mi+1,...,mj−1,...,mp)
contradicting the choice of Km1,...,mi,...,mj ,...,mp . Hence, all parameters m1, . . . ,mp are
equal to one, except for one parameter equal to n−p+1, so thatKm1,...,mp ∼= CS(n, p−1).

Proof of Theorem 1.7 Let Km1,...,mp be the complete multipartite graph with the largest
spectral radius. It is apparent from Lemma 2.1, that |mi −mj | ≤ 1 holds for all i 6= j, as
otherwise, assuming mi −mj ≥ 2,
λ1(Km1,...,mi−1,...,mj+1,...,mp) > λ1(Km1,...,mi,...,mj ,...,mp)
contradicting the choice of Km1,...,mi,...,mj ,...,mp . The condition |mi −mj | ≤ 1 for i 6= j
implies that each parameter mi is equal to either bn/pc or dn/pe, so that Km1,...,mp ∼=
T (n, p). 
Remark 2.2. Delorme [5] proved (a bit too concisely) that changing the arbitrary e param-
eters of a complete multipartite graph by their average value increases the spectral radius by
relying on the characteristic polynomial. While this result can be substituted for Lemma 2.1
in the proofs of Theorems 1.6 and 1.7, Lemma 2.1 is an independent result based on the
principal eigenvector and inspired by the rotation lemma from [1].
Remark 2.3. Delorme also asked in [5] whether the spectral radius of complete multi-
partite graph Kn1,...,np is a concave function on the (p − 1)-dimensional simplex Y :∑p
i=1 ni = n ∧ (∀i ∈ {1, . . . , p})(ni ≥ 0), i.e., whether
λ1(Kt(n1,...,np)+(1−t)(m1,...,mp)) ≥ t λ1(Kn1,...,np) + (1− t)λ1(Km1,...,mp)
for any two points (n1, . . . , np), (m1, . . . ,mp) ∈ Y and each t ∈ [0, 1] such that
t(n1, . . . , np) + (1− t)(m1, . . . ,mp) ∈ Y ?
Delorme proved this affirmatively for p ≤ 3 in [5]. We tested it computationally for t =
0.5, p ∈ {4, . . . , 10} and n ≤ 33 and found no counterexamples to the above question on
these simplices.
D. Stevanović et al.: On spectral radius and energy of complete multipartite graphs 113
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