Graphs with maximal Laplacian energy

Dragan Stevanovic1

FAMNIT—University of Primorska, Glagoljaska 8, 6000 Koper, Slovenia, and Faculty ofSciences and Mathematics, University of Nis, Visegradska 33, 18000 Nis, Serbia

Abstract

Maximum Laplacian Energy . . .

Keywords: Laplacian energy, Threshold graphs, Trees, Unicyclic graphs2010 MSC: 05C50

1. Introduction

Let G = (V,E) be a simple undirected graph with n = |V | vertices andm = |E| edges. Let L = D − A be the Laplacian matrix of G, where D =diag(d1, . . . , dn) is the diagonal matrix of vertex degrees and A is the adjacencymatrix of G. Further, let µ1 ≥ . . . µn−1 ≥ µn = 0 be the eigenvalues of L. Forconnected G one has µn−1 > 0 [3].

The Laplacian energy has been defined in [5] as

LE(G) =

n∑i=1

∣∣∣∣µi − 2m

n

∣∣∣∣ .It has been conjectured there that, among trees, the path has minimum Lapla-cian energy and the star has maximum Laplacian energy. The second part ofthis conjecture has been recently proved in [4].

AIM workshop in 2006Trevisan’s recent paperGoalsWe are going to base our proofs on three significant results: the Koolen-

Moulton bound on the energy of graphs, Byer’s description of graphs withmaximum sum of squared vertex degrees and Nikiforov’s bounds on the sumof squared vertex degrees in graphs.

Paper organization

Email address: dragance106@yahoo.com (Dragan Stevanovic)1The author was supported by Grant 174033 of Serbian Ministry of Science and Techno-

logical Development and Research Programme P1-0285 of Slovenian Research Agency.

Preprint submitted to Elsevier April 12, 2011

2. Koolen-Moulton approach

We haven−1∑i=1

µi = Tr(L) = 2m, (1)

andn−1∑i=1

µ2i = Tr(L2) =

n∑i=1

di(di + 1) = 2m+M, (2)

where M =∑ni=1 d

2i is the sum of squares of the vertex degrees (also known as

the first Zagreb index [8]). From (1) we at once get µ1 ≥ 1n−1

∑n−1i=1 µi = 2m

n−1 .It also holds that µ1 ≤ n [9].

Suppose for the moment that n, m and M are fixed. Since the smallestLaplacian eigenvalue µn = 0 is constant, the term

∣∣0− 2mn

∣∣ = 2mn is always

present as a summand in LE(G), and so we have

LE(G) =

(µ1 −

2m

n

)+

n−1∑i=2

∣∣∣∣µi − 2m

n

∣∣∣∣+2m

n

= µ1 +

n−1∑i=2

∣∣∣∣µi − 2m

n

∣∣∣∣ .We may use the Cauchy-Schwartz inequality to estimate the above sum

n−1∑i=2

∣∣∣∣µi − 2m

n

∣∣∣∣ ≤√√√√n−1∑

i=2

12 ·n−1∑i=2

(µi −

2m

n

)2

=

√√√√(n− 2)

n−1∑i=2

(µ2i −

4mµin

+4m2

n2

)

=

√√√√(n− 2)

(n−1∑i=2

µ2i −

4m

n

n−1∑i=2

µi + (n− 2)4m2

n2

)

=

√(n− 2)(2m+M − µ2

1)− 4m(n− 2)

n(2m− µ1) +

4m2(n− 2)2

n2.

Therefore,

LE(G) ≤ µ1+

√(n− 2)(2m+M − µ2

1)− 4m(n− 2)

n(2m− µ1) +

4m2(n− 2)2

n2= F (µ1,M).

The partial derivatives of F (µ1) are equal to

∂F

∂M=

n− 2

2√

(n− 2)(2m+M − µ21)− 4m(n−2)

n (2m− µ1) + 4m2(n−2)2n2

,

∂F

∂µ1= 1−

(n− 2)(µ1 − 2m

n

)√(n− 2)(2m+M − µ2

1)− 4m(n−2)n (2m− µ1) + 4m2(n−2)2

n2

.

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Thus, ∂F∂M > 0 and so, for fixed µ1, the function F (µ1,M) increases with M .

On the other hand, from the condition F ′(µ1) = 0 we get

(n−2)(2m+M−µ21)−4m(n− 2)

n(2m−µ1)+

4m2(n− 2)2

n2= (n−2)2

(µ1 −

2m

n

)2

,

i.e.,

(n− 1)µ21 −

4m(n− 1)

nµ1 +

(8m2

n− 2m−M

)= 0,

which is a quadratic equation in µ1, whose solutions are

µ1 =2m

n±

√2m+M

n− 1− 4m2(n+ 1)

n2(n− 1).

Since ∂F∂µ1

(2mn

)= 1 > 0, for fixed M , the function F increases on [ 2mn ,

2mn +√

2m+Mn−1 −

4m2(n+1)n2(n−1) ] and then starts to decrease. Thus, for fixed M , the max-

imum of F (µ1,M) on the interval µ1 ∈ [ 2mn , n] is reached either at µ1 = n if

n ≤ 2mn +

√2m+Mn−1 −

4m2(n+1)n2(n−1) , in which case

LE(G) ≤ n+

√M(n− 2)− n2(n− 2)− 4m2 − 12m+ 6mn+

16m2

n2= G1(M),

or at µ1 = 2mn +

√2m+Mn−1 −

4m2(n+1)n2(n−1) if n > 2m

n +√

2m+Mn−1 −

4m2(n+1)n2(n−1) , in which

case

LE(G) ≤ 2m

n+

√M

n− 1+ sn,m +

√M(n− 2)2

n− 1+ tn,m = G2(M),

for sn,m = 2mn−1 − 4m2 n+1

n2(n−1) and tn,m = 2m (n−2)2n−1 − 4m2 (n−2)(n2−n−2)

(n−1)n2 .

The big question: when is n ≤ 2mn +

√2m+Mn−1 −

4m2(n+1)n2(n−1) ? In that case the

maximum of F is clear: take µ1 = n and maximum possible M .

n− 2m

n≤

√2m+M

n− 1− 4m2(n+ 1)

n2(n− 1)(n− 2m

n

)2

≤ 2m+M

n− 1− 4m2(n+ 1)

n2(n− 1)

n2 +4m2

n2− 4m+

4m2(n+ 1)

n2(n− 1)≤ 2m+M

n− 1

n2 − 4m+8m2

n(n− 1)≤ 2m+M

n− 1

n2(n− 1)− 4m(n− 1) +8m2

n≤ 2m+M

n2(n− 1)− 4m(n− 1)− 2m+8m2

n≤M.

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On the other hand, we already know that M ≤ 2m√

2m for m ≥ n2/4 [10],so the above inequality may hold only if

n2(n− 1)− 2m(2n− 1) +8m2

n≤ 2m

√2m

n2(n− 1) +8m2

n≤ 2m

(2n− 1 +

√2m)

This will hold for large values of m, but I do not know how large. The other Mbound from [10], the one for m < n2/4, will probably yield that no m satisfiesthe bound (this should be checked).

Therefore, for large m the situation will be crystally clear: take µ1 = nand maximum M , check out Byer’s constructions and pick the winner. Here Ineed an excellent estimate for m. The conjectured maximum pineapple is outof reach, as it has m ≈ 2n2/9. Moreover, you can conclude that the graph hasmaximum LE only if it further satisfies the case of equality in the initial Cauchy-Schwartz inequality. Altogether, this may give some maximal LE graphs, butnot the total extreme.

Otherwise, when n > 2mn +

√2m+Mn−1 −

4m2(n+1)n2(n−1) , the situation is more com-

plicated: the maximum of F is found along the northern border of the set offeasible pairs (µ1,M) for all graphs with n vertices and m edges.

Thus, for any fixed value of µ1, one should take maximum feasible M as afunction of µ1, and then optimize over all µ1.

Here one could first derive some bounds for M depending on µ1, with similarapproach. For example, (

n−1∑i=2

µi

)2

≤n−1∑i=2

12 ·n−1∑i=2

µ2i

(2m− µ1)2 ≤ (n− 2)(2m+M − µ21)

(2m− µ1)2

n− 2− 2m+ µ2

1 ≤ M.

Perhaps I would have to consider explicitly the maximum of M when themaximum vertex degree is bounded, as it obviously holds µ1 ≥ ∆ + 1, andtherefore, for fixed µ1, we have ∆ ≤ µ1−1. This would have to further improveByer’s 1999 paper, and it might be a tremendous job, as it takes on anotherhuge task.

Anyway, I need to know the northern border of the defining area in orderto find the maximum of F . Perhaps it may be done more easily for trees—Mshould behave simpler then, and the maximum will probably be obtained forcomplete ∆-ary trees.

Otherwise, things may become easier if we just fix n and optimize also form. Perhaps in that case I won’t need to know the northern border exactly, butcould use just one extreme point for each m.

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Bound developmentArgument for upper bound maximizationMmax, µmax graphsit holds if m ≤ 0.155n2 and large enough nDeriving an upper bound for all n, m(≤ 0.155n2)

3. Bound is sharp for

Overview of Byer’s graphs and their spectraEquality condition satisfied for

3.1. m = n− 1, trees

The bound could be reworked to remove the condition for large enough n.

3.2. m = n, unicyclic graphs

Star with an extra edge. Reworking the bound?

3.3. Pineaple PAn,n

A very special case indeed!

4. Further observations

Drawings of maximal LE graphs for either n = 9 or n = 10, and all feasiblem

Observation that each such graph is a threshold graphImportant: however, not every max graph is one of Byer’s graphs!Conjecture on maximum LE for fixed n

5. Perhaps for another paper

Write, then send to Simic and Ilic for discussion.Calculation of Laplacian spectrum and LE for threshold graphs, then which

threshold graph has max LE. Formula?The proof of conjecture among threshold graphsWhat remains is to explain why threshold graphs appear here? Apparently,

it is not to maximize M and Koolen-Moulton approach may no longer be used.

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References

[1] O.D. Byer, Two path extremal graphs and an application to a Ramsey-typeproblem, Discrete Math. 196 (1999), 51–64.

[2] D. Cvetkovic, M. Doob, H. Sachs, Spectra of graphs—Theory and Applica-tion, 3rd edition, Johann Ambrosius Barth Verlag, 1995.

[3] M. Fiedler, classic—algebraic connectivity is nonzero in connected graphs.

[4] E. Fritscher, C. Hoppen, I. Rocha, V. Trevisan, On the sum of the Laplacianeigenvalues of a tree, Linear Algebra Appl. 435 (2011), 371–399.

[5] I. Gutman, B. Zhou, Laplacian energy introduction and definition, LinearAlgebra Appl. 2006.

[6] J. Koolen, V. Moulton, the first paper with upper bound for general graphs.

[7] J. Koolen, V. Moulton, the second paper with upper bound for bipartitegraphs.

[8] Some reference/survey to the first Zagreb index.

[9] Some reference/survey to the Laplacian spectral radius.

[10] V. Nikiforov, The sum of the squares of degrees: Sharp asymptotics, Dis-crete Math. 307 (2007), 3187–3193.

[11] D. Stevanovic, link to my slides from the AIM workshop, 2006.

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Table 1: Connected graphs with maximum Laplacian energy with nine vertices and 8 ≤ m ≤36 edges

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