Open problems on graph eigenvalues studied with

AutoGraphiX

Mustapha Aouchiche, Gilles Caporossi and Pierre Hansen

GERAD and HEC Montreal, Canada

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PLAN

1 AutoGraphiX (AGX)

2 Variable neighborhood search (VNS)

3 Conjecture generation4 Applications :

The irregularityThe spectral spreadNordhaus–Gaddum inequality for the indexThe Gutman energy

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1. AutoGraphiX (AGX)

AGX: The basic idea

The research problem that led to AutoGraphiX (G. Caporossi, P. Hansen 1998,1999, 2000,...) is based on the following two observations:

Many problems in extremal graph theory can be formulated as parametriccombinatorial optimization problem on an infinite family of graphs (ofwhich only those of moderate size are considered in practice).

These problems can be solved approximately using a generic heuristic(instead of requiring, as in Operations Research, specific algorithms foreach problem).

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1. AutoGraphiX (AGX)

AGX: Main tasks

Search for graphs subject to constraints

Search for extremal graphs (that maximize or minimize a given invariant)

Corroborate, strengthen, refute or find conjectures in graph theory(numerical, geometric and algebraic methods to find conjectures)

Find ideas of proofs

Find automated proofs of simple results

In addition, AGX offers a convivial interactive mode to PLAY graph theoryPLAY = display and modify (edge and vertex addition/removal), displayinformation (values, edge/vertex important sebsets) about the graph

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1. AutoGraphiX (AGX)

AGX: Principle

To do this, AutoGraphiX exploits the variable neighborhood searchmetaheuristic to solve approximately problems of the form:

i : graph invariant (quantity that doesn’t change underisomorphism of G ), possibly an algebraic formula involving manyinvariantsGn : family of graphs of order n

Gn,m : family of graphs of order n and size m

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2. Variable neighborhood search (VNS)

VNS: A basic tool

A metaheuristic is a general framework for building heuristics applied invarious combinatorial or global optimization problems.

The variable neighborhood search metaheuristic is due to N. Mladenovićand P. Hansen (1997).

It exploits a systematic change in the neighborhood in a descent phasetowards a local optimum as well as in a perturbation phase in order to getout of the corresponding valley.

This metaheuristic has been widely used in Operations Research, inArtificial Intelligence and in Data Mining (see Google Scholar for over onethousand citations).

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2. Variable neighborhood search (VNS)

VNS: A basic tool

VNS iterations

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2. Variable neighborhood search (VNS)

VNS: A basic tool

The VNS heuristic

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1. Introduction to AutoGraphiX

VNS: A basic tool

Some basic moves used to define neighborhoods in graph theory

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3. Conjecture generation

Numerical method

Search for extremal graphs related to a function of invariants(Parametric approximate optimization using AutoGraphiX)

Consider these graphs as points in the invariant space

Apply a PCA based method (included in AutoGraphiX) to findlinear relations, if any, between selected invariants

Each relation corresponds to a conjecture(lower bound if the objective is minimized; upper bound if the objective is

maximized)

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3. Conjecture generation

Algebraic method

Generate extremal graphs using a parametrization on one ormore invariants

Recognize the family (or families) of extremal graphs

Find conjectures by substituting to the invariants theirexpressions as functions of the order n on the family (orfamilies) of extremal graphs recognized

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3. Conjecture generation

Geometric method

Generate extremal graphs using a parametrization on one ormore invariants

Consider these graphs as points in the invariant space

Find the convex hull of the point set

Each facet corresponds to a conjecture, in the form of aninequality between invariants

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4. Applications

Irregularity

In 1957, Collatz & Sinogowitz

proved: λ1(G ) ≥ d(G ) with equlity iff G is regular

proposed a measure of irregularity: Ir(G ) = λ1(G )− d(G )

λ1(G ): the index (spectral radius) of (the adjacency matrix of) G

d(G ): the average degree of G

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4. Applications

Irregularity

Extremal graphs obtained by AGX

A pineapple PA(n, q) on n vertices consists of a clique Kq and anindependent set S on n − q vertices in which each vertex of S isadjacent to a unique and the same vertex of Kq.

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4. Applications

Irregularity

The observation of the extremal graphs (algebraic methode) led toa structural conjecture

Conjecture 1: [Aouchiche et al. European J. Oper. Res. 2008]The most irregular connected graph on n (n ≥ 10) vertices is apineapple PA(n, q) in which the clique size q is equal to ⌈n

2⌉+ 1.

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4. Applications

Irregularity

Some partial results:

Proposition: [Aouchiche et al. EJOR 2008]For any connected graph G on n vertices

λ1(G )− d(G ) ≤ n

4− 1 +

1

n.

Proposition:

For an integer n ≥ 10, let bn = max{irr(G ) : G connected and |G | = n}.Then

bn >

n

4− 3

2+ 2

nif n is even;

n

4− 3

2+ 7

4nif n is odd.

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4. Applications

Irregularity

The minimum degree of the most irregular graph is δ = 1

Theorem:

Let G be a connected graph on n vertices with minimum degreeδ ≥ 2. Then

λ1(G )− d(G ) ≤ n

4− 3

2+

9

4n< λ1(PA(n, q))− d(PA(n, q)),

where q =⌈

n

2

⌉

+ 1.

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4. Applications

Spectral spread

The spectral spread of a graph G is the difference between thelargest and the least of its eigenvalues

s(G ) = λ1(G )− λn(G )

A complete split graph CS(n, q) on n vertices consists of a cliqueKq and an independent S on n − q vertices in which each vertex ofKq is adjacent to each vertex of S

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4. Applications

Spectral spread

AGX suggests that the complete split graphs maximize the spectralspread

Conjecture 2: (see also [Aouchiche et al. EJOR 2008]Let G be a connected graph on n ≥ 3 vertices. Then

s(G ) ≤√

4qn − 3q2 − 2q + 1

with equality iff G is the complete split graph CS(n, q) with anindependent set of size n − q =

⌈

n

3

⌉

and a clique of size q =⌊

2n3

⌋

.

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4. Applications

Nordhaus–Gaddum inequality

Theorem: [Nordhaus & Gaddum, Amer. Math. Monthly 1956]Let G be a graph and G its complement. If χ denotes thechromatic number, then

2√

n ≤ χ(G ) + χ(G ) ≤ n + 1 and n ≤ χ(G ) · χ(G ) ≤ (n + 1)2

4

Since 1956, many Nordhaus-Gaddum type inequalities were provedfor different invariants

Among these invariants the spectral radius

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4. Applications

Nordhaus–Gaddum inequality

Some known results:

√

2n(n − 1)− 4δ(n − 1 −∆) + 1 − 1 [Li, 1996]√

(

2 −1

χ(G)−

1χ(G)

)

n(n − 1) [Hong & Shu, 2000]

√

(

2 −1

θ(G)−

1θ(G)

)

n(n − 1) [Nikiforov, 2002]

√

2 ((n − 1)2 − 2δn + 2∆δ −∆+ 3δ) [Shi, 2007]

n−∆+δ−3+√

2((n−∆)2+4n(∆−δ)+(δ+1)2)

2[Shi, 2007]

where δ, ∆, χ and θ denote the minimum degree, maximum degree, chromaticnumber and maximum clique number, respectively.

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4. Applications

Nordhaus–Gaddum inequality

AGX results:

The extremal graphs are complete split graphs with q = ⌊n/3⌋

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4. Applications

Nordhaus–Gaddum inequality

Best known bound:

λ1(G) + λ1(G) ≤ 1 +√

3

2n − 1 [Csikvári 2009]

AGX suggests:

Conjecture 3:

For any simple graph G on n vertices with complement G , we have

λ1(G) + λ1(G) ≤ 4

3n − 5

3−

3n−2−√

9n2−12n+12

6if n mod(3) = 1

0 if n mod(3) = 23n−1−

√9n2

−6n+9

6if n mod(3) = 0.

The bound is sharp as shown by the complete split graphs with q = ⌊n/3⌋.

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4. Applications

Nordhaus–Gaddum inequality

Partial results: [Aouchiche et al. EJOR 2008]

Let G be a connected graph on n vertices with minimum degree δ andmaximum degree ∆.

If ∆− δ ≤ n−23

, then

λ1(G) + λ1(G ) ≤ 4

3n − 5

3.

If G is a complete split graph CS(n, q), then

λ1(G) + λ1(G) ≤ 4

3n − 5

3−

3n−2−√

9n2−12n+12

6if n mod(3) = 1

0 if n mod(3) = 23n−1−

√9n2

−6n+9

6if n mod(3) = 0

with equality iff q =⌊

n3

⌋

for all n or q =⌈

n3

⌉

for n mod(3) = 2.

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4. Applications

Gutman energy

The energy E (G) of a graph G [Gutman, 1978] is the sum of the absolutevalues of its eigenvalues

E (G) =

n∑

i=1

|λi (G)| = 2∑

λi>0

λi (G) = 2∑

λi<0

|λi (G)|.

AGX conjectured lower bounds

Theorem: [Caporossi et al. J. Chem. Inf. Comp. Sc. 1999]Let G be a simple graph on n vertices and m edges with energy E . Then

1 E ≥ 4m/n;

2 E ≥ 2√

m with equality iff G is a complete bipartite graph plus possiblysome isolated vertices;

3 if G is connected, E ≥ 2√

n − 1 with equality iff G is the star Sn;

4 E ≤ 2m with equality iff G is composed of disjoint edges and possiblyisolated vertices.

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4. Applications

Gutman energy

Maximizing E (G) for a connected k–cyclic graph (a graph that containsexactly k cycles) remains an open problem

A lollipop Loln,g is a graph obtained from a cycle Cg and a path Pn−g byadding an edge between a vertex from the cycle and an endpoint from the path

Using AGX, Caporossi et al. [J. Chem. Inf. Comp. Sci. 1999] suggested

Conjecture 4:

Among unicyclic graphs on n vertices the cycle Cn has maximal energy if n ≤ 7and n = 9, 10, 11, 13 and 15. For all other values of n the unicyclic graph withmaximum energy is the lollipop Loln,6.

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4. Applications

Gutman energy

Conjecture 4 was widely studied and a series of related results were proved

Theorem: [Hou, Gutman and Woo, 2002] Let G be a connected, unicyclic andbipartite graph on n ≥ 7 vertices and G 6∼= Cn. Then E (G) ≤ E (Loln,6).

Theorem: [Huo, Li and Shi, LAA 2011] For n = 8, 12, 14 and n ≥ 16, we haveE (Loln,6) > E (Cn).

Theorem: [Huo, Li and Shi, Euro. J. Comb. 2011] Among all unicyclic graphson n vertices, the cycle Cn has maximal energy if n ≤ 7 but n 6= 4, andn = 9, 10, 11, 13 and 15; Lol4,3 has maximal energy if n = 4. For all othervalues of n, the unicyclic graph with maximal energy is Loln,6.

Theorem: [Andriantiana and Wagner, LAA 2011] Among all non-bipartiteunicyclic graphs on n ≥ 3 vertices, Loln,3 has maximum energy if n is even, andCn has maximum energy if n is odd.

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4. Applications

Gutman energy

The case of bicyclic graphs

Conjecture: [Gutman and Vidović, J. Chem. Inf. Sci. 2001] For n = 14 andn ≥ 16, the bicyclic molecular graph of order n with maximal energy is themolecular graph of the α, β diphenyl–polyene C6H5(CH)n−12C6H5, which isrepresented by the graph P6;6

n obtained from two copies of the cycle C6 and apath Pn−12 by attaching with an edge each endpoint of the path to one cycle.

Theorem: [Huo, Ji, Li and Shi, LAA 2011] Let G be any connected, bipartitebicyclic graph on n ≥ 12 vertices. Then E (G) ≤ E (P6;6

n ) with equality if andonly if G ≡ P6;6

n .

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4. Applications

Gutman energy

The case of tricyclic and quadricyclic graphs

Conjecture 4: Let G be a tricyclic graph on n vertices with n = 20 or n ≥ 22.Then E (G) ≤ E (P6,6;6

n ) with equality if and only if G ≡ P6,6;6n .

Conjecture 5: Let G be a quadricyclic graph on n vertices with n ≥ 26. ThenE (G) ≤ E (P6,6;6,6

n ) with equality if and only if G ≡ P6,6;6,6n .

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4. Applications

Gutman energy

The general case

For any integer n, p and q such that n ≥ 6k + 2 with k = p + q, let Pp×6;q×6n

be the graph obtained from k copies of C6 and a path Pn−6k with endpoints u

and v , by adding an edge between u and each of p copies of C6 and an edgebetween v and each of the q other copies of C6.

Conjecture 6:

Let k and n be positive integers such that n ≥ 6k + 4. Then for any connectedk–cyclic graph G , E (G) ≤ E (Pp×6;q×6

n ), where p = ⌊k/2⌉ and q = ⌊k/2⌋, withequality if and only if G ≡ Pp×6;q×6

n .

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