Resolvent Energy of Graphs - MISANU · Resolvent Energy of Graphs I.Gutman1;2, B.Furtula1, E.Zogi c2, E.Glogi c2 1 Faculty of Science, University of Kragujevac, Kragujevac, Serbia

Resolvent Energy of Graphs

I.Gutman1,2, B.Furtula1, E.Zogic2, E.Glogic2

1 Faculty of Science, University of Kragujevac, Kragujevac, Serbia2 State University of Novi Pazar, Novi Pazar, Serbia

May 19, 2016

I.Gutman1,2, B.Furtula1, E.Zogic2, E.Glogic2 ER(G) May 19, 2016 1 / 25

Introduction

I. Gutman, B. Furtula, E. Zogic, E. Glogic, Resolvent energy of graphs,MATCH Communications in Mathematical and in Computer Chemistry 75(2016) 279 - 290.


Introduction

M-square matrix of order n; Resolvent matrix of M:

RM(z) = (zIn −M)−1 (1)

RM(z) exists for all values of z except when z coincides with aneigenvalue of M.

G -simple graph; A-adjacency matrix; λ1, λ2, ..., λn eigenvalues

The k-th spectral moment of G :

Mk(G ) =n∑

i=1

(λi )k (2)

M0 = n,M1 = 0,M2 = 2m, Mk = 0 for all odd values of k if and onlyif G is bipartite.

1 D. Cvetkovic, M. Doob, H. Sachs, Spectra of Graphs Theory and Application, 1980.

2 T. S. Shores, Linear Algebra and Matrix Analysis, 2007.


Introduction


RM(z) = (zIn −M)−1 (1)




Mk(G ) =n∑

i=1

(λi )k (2)





Introduction


RM(z) = (zIn −M)−1 (1)




Mk(G ) =n∑

i=1

(λi )k (2)





Introduction


RM(z) = (zIn −M)−1 (1)




Mk(G ) =n∑

i=1

(λi )k (2)





Introduction

Energy of graph G

E(G) =n∑

i=1

|λi | (3)

1 I. Gutman, The energy of a graph, (1978)2 X. Li. Y. Shi, I. Gutman, Graph Energy, (2012)3 I. Gutman, X. Li (Eds.), Energies of Graphs-Theory and Applications, (2016)

Resolvent matrix of adjency matrix A

RA(z) = (zIn − A)−1 (4)

1

z − λi, i = 1, 2, ..., n − eigenvalues of RA(z) (5)

λi ≤ n − 1 is satisfied by all eigenvalues of all n-vertex graphs.

Resolvent energy of graph G

ER(G) =n∑

i=1

1

n − λi(6)


Introduction

Energy of graph G

E(G) =n∑

i=1

|λi | (3)



RA(z) = (zIn − A)−1 (4)

1




ER(G) =n∑

i=1

1

n − λi(6)


Introduction

Energy of graph G

E(G) =n∑

i=1

|λi | (3)



RA(z) = (zIn − A)−1 (4)

1




ER(G) =n∑

i=1

1

n − λi(6)


Introduction

Resolvent Estrada index

EEr (G ) =∞∑k=0

Mk(G )

(n − 1)k(7)

EEr (G ) =n∑

i=1

n − 1

n − 1− λi(8)

EEr is undefined in the case of the complete graph Kn.

X. Chen, J. Qian, Bounding the resolvent Estrada index of a graph, (2012)

X. Chen, J. Qian, On resolvent Estrada index, (2015)

I. Gutman, B. Furtula, X. Chen, J. Qian, Graphs with smallest resolvent Estrada indices(2015)

I. Gutman, B. Furtula, X. Chen, J. Qian, Resolvent Estrada index - computational andmathematical studies, (2015)


Basic properties of resolvent energy

Theorem 1.

If G is graph of order n and Mk(G ), k = 0, 1, 2, ... its spectral moments,then

ER(G ) =1

n

∞∑k=0

Mk(G )

nk. (9)

Corollary 1.

If e is a edge of the graph G , denote by G − e the subgraph obtained bydeleting e from G . Then for any edge e of G ,

ER(G − e) < ER(G ).



Theorem 1.

If G is graph of order n and Mk(G ), k = 0, 1, 2, ... its spectral moments,then

ER(G ) =1

n

∞∑k=0

Mk(G )

nk. (9)

Corollary 1.

If e is a edge of the graph G , denote by G − e the subgraph obtained bydeleting e from G . Then for any edge e of G ,

ER(G − e) < ER(G ).



Corollary 2.

Let Kn be the complete graph of order n and Kn the edgeless graph oforder n. Then for any graph G of order n, different from Kn and Kn,

ER(Kn) < ER(G ) < ER(Kn), (10)

1 ≤ ER(G ) ≤ 2n

n + 1. (11)

ER(G ) = 1 if and only if G ∼= Kn while ER(G ) = 2nn+1 if and only if

G ∼= Kn.



Corollary 3.

Among connected graphs of order n, the graph with the smallest resolventenergy is a tree.

Theorem 2.

Amog trees of order n, the path Pn has the smallest and the star Sn hasthe greatest resolvent energy.

H. Deng, A proof of a conjecture on the Estrada index, (2009)

M2k (Pn) ≤ M2k (T ) ≤ M2k (Sn)

M2k (Pn) < M2k (T ) < M2k (Sn), for k ≥ 2

ER(G) =1

n

∞∑k=0

Mk (G)

nk



Corollary 3.

Among connected graphs of order n, the graph with the smallest resolventenergy is a tree.

Theorem 2.

Amog trees of order n, the path Pn has the smallest and the star Sn hasthe greatest resolvent energy.

H. Deng, A proof of a conjecture on the Estrada index, (2009)

M2k (Pn) ≤ M2k (T ) ≤ M2k (Sn)

M2k (Pn) < M2k (T ) < M2k (Sn), for k ≥ 2

ER(G) =1

n

∞∑k=0

Mk (G)

nk



Theorem 3.

Let G be a graph of order n and φ(G , λ) = det(λIn − A(G )) ne itscharacteristic polynomial. Then

ER(G ) =φ′(G , n)

φ(G , n), (12)

where φ′(G , λ) is the first derivative of φ(G , λ).


Estimating the resolvent energy

Theorem 4.

Let G be a graph with n vertices and m edges. Then

n3

n3 − 2m≤ ER(G ) ≤ 1 +

2m(2n − 1)

n2(n2 − 2m). (13)

Equality on the left-hand side is attained if and only if G ∼= Kn or(provided n as even) G ∼= n

2K2. Equality on the right-hand side is attainedif and only if G ∼= Kn.



Proposition 1.

Let G be a graph with n vetrices, m edges, and nullity n0. Then

ER(G ) ≥ n0

n+

n(n − n0)2

n2(n − n0)− 2m. (14)

Proposition 2.

If G is bipartite graph with n vertices and m edges where n is odd number,then n0 ≥ 1 and

ER(G ) ≥ 1

n+

n(n − 1)2

n2(n − 1)− 2m. (15)

D. Cvetkovic, I. Gutman, The algebraic multiplicity of the number zero in the spectrum ofa bipartite graph, (1972)



Proposition 3.

Let G be a bipartite graph with n vertices and m edges. Then

ER(G ) ≤ 1 +2m

n(n2 −m). (16)

Equality is attained if and only if either G ∼= Kn or G ∼= Ka,b ∪ Kn−a−b,where Ka,b is a complete bipartite graph such that ab = m.


Computational studies on resolvent energy

The ER-values of all trees and connected unicyclic, and bicyclic graphs up to 15 vertices werecomputed, and the structure of the extremal members of these classes was established.First of all, the results of Theorem 2 were confirmed and extended:

Observation 1.Among trees of order n, the path Pn has the smallest and the tree P∗n second-smallest resolventenergy. Among trees of order n, the star Sn has the greatest and the tree S∗n second-greatestresolvent energy.



Observation 2.

Among connected unicyclic graphs of order n, n ≥ 4, the cycle Cn has thesmallest and the graph C ∗n second-smallest reslovent energy. Among thesegraphs of order n, n ≥ 5, the graphs Xn and X ∗n have, respectively, thegreatest and second-greatest resolvent energy.



Obsevation 3. Among connected bicyclic graphs of order n, those with the smallest, (a),and second-smallest, (b), resolvent energy are:

(a)Bp−1,p−1,p if n = 3p, p ≥ 2Bp−1,p,p if n = 3p + 1, p ≥ 1Bp,p,p if n = 3p + 2, p ≥ 1.

(b)Bp−2,p,p if n = 3p, p ≥ 2Bp−1,p−1,p+1 if n = 3p + 1, p ≥ 2Bp−1,p,p+1 if n = 3p + 2, p ≥ 1.

Among these graphs of order n, n ≥ 5, the graph Yn has the greatest resolvent energy. Forn ≥ 9, the graph Y ∗n has second-greatest resolvent energy, whereas Y ∗5 ,Y

∗6 ,Y

∗7 and Y ∗8 are

exceptions.



Observation 4.

Among connected tricyclic graphs of order n, n ≥ 5, the graph Zn has thegreatest resolvent energy. For n ≥ 6, graph Z ∗n has second-greatest energy,whereas Z ∗5 is exception.



Observation 5.

Connected tricyclic graphs of order n, 5 ≤ n ≤ 15 with the smallest ER,are depicted in the following figure.



Observation 6.

The inequality ER(Sn) < ER(Cn) holds for all n ≥ 4. Consequently,any tree has smaller ER-value than any unicyclic graph of the sameorder.

For Ba,b,c specified by (a) from Observation 3, the inequalityER(Xn) < ER(Ba,b,c) holds only until n = 6 and is violated for alln ≥ 7. Consequently, it is not true that any unicyclic graph hassmaller ER-value than any bicyclic graph of the same order.

The same applies also to the relation between ER of bicyclic andtricyclic graphs.

Any unicyclic graph has smaller ER-value than any connected tricyclicgraph of the same order.



Observation 7.

Cospectral graphs have equal ER-values.

There are non-cospectral graphs whose ER-values are different, butremarkably close. ER(B3,3,3) = 1.018571022 whereasER(B2,3,4) = 1.018571080, and ER(B4,4,5) = 1.0096261837436whereas ER(B3,5,5) = 1.0096261837458.


References

M. Benzi, P. BoitoQuadrature rule-based bounds for functions of adjacency matricesLin. Algebra Appl. 433 (2010) 637-652.

S. B. Bozkurt, A. D. Gungor, I. GutmanRandic spectral radius and Randic energy,MATCH Commun. Math. Comput. Chem. 64 (2010) 321-334.

S. B. Bozkurt, A. D. Gungor, I. Gutman, A. S Cevik,Randic matrix and Randic energyMATCH Commun. Math. Comput. Chem. 64 (2010) 239-250.

X. Chen, J. Qian,Bounding the resolvent Estrada index of a graph,J. Math. Study 45 (2012) 159-166.

X. Chen, J. Qian,On resolvent Estrada index,MATCH Commun. Math. Comput. Chem. 73 (2015) 163-174.

D. M. Cvetkovic, I. Gutman,The algebraic multiplicity of the number zero in the spectrum of a bipartite graph,Mat. Vesnik (Beograd) 9 (1972) 141-150.


References

D. Cvetkovic, P. Rowlinson, S. Simic,An Introduction to the Theory of Graph Spectra,Cambridge Univ. Press, Cambridge, 2010.

D. Cvetkovic, M. Doob, H. Sachs,Spectra of Graphs-Theory and Application,Academic Press, New York, 1980.

K. C. Das, S. Sorgun, K. Xu,On Randic energy of graphs, in: I. Gutman, X. Li (Eds.), Energies of Graphs-Theory andApplications,Univ. Kragujevac, Kragujevac, 2016, pp. 111-122.

H. Deng,A proof of a conjecture on the Estrada index,MATCH Commun. Math. Comput. Chem. 62 (2009) 599-606.

I. Gutman,The energy of a graph,Ber. Math.-Statist. Sekt. Forschungsz. Graz. 103 (1978) 1-22.

I. Gutman,Comparative studies of graph energies,Bull. Acad. Serbe Sci. Arts (Cl.Sci. Math. Nature.) 144 (2012) 1-17.


References

I. Gutman,Census of graph energies,MATCH Commun. Math. Comput. Chem. 74 (2015) 219-221.

I. Gutman,B. Furtula, X. Chen, J. Qian,Graphs with smallest resolvent Estrada indices,MATCH Commun. Math. Comput. Chem. 73 (2015) 267-270.

I. Gutman,B. Furtula, X. Chen, J. Qian,Resolvent Estrada index-computational and mathematical studies,MATCH Commun. Math. Comput. Chem. 74 (2015) 431-440.

I. Gutman, B. Furtula, E. Zogic, E. Glogic,Resolvent energy, in: I. Gutman, X. Li (Eds.), Graph Energies-Theory and Applications,Univ. Kragujevac, Kragujevac, 2016, pp. 277-290.

I. Gutman, X. Li (Eds.),Energies of Graphs-Theory and Applications,Univ. Kragujevac, Kragujevac, 2016.

E. Estrada, D. J. Higham,Network properties revealed through matrix functions,SIAM Rev. 52 (2010) 696-714.


References

J. Li, J. M. Guo, W. C. Shiu,A note on Randic energy,MATCH Commun. Math. Comput. Chem. 74 (2015) 389-398.

X. Li. Y. Shi, I. Gutman,Graph Energy,Springer, New York, 2012.

O. Miljkovic, B. Furtula, S. Radenkovic, I. Gutman,Equienergetic and almost-equienergetic trees,MATCH Commun, Math. Comput. Chem. 61 (2009) 451-461.

D. S. Mitrinovic, P. Vasic,Analytic Inequalities,Springer, Berlin, 1970.

V. Nikiforov,The energy of graphs and matrices,J. Math. Anal. Appl. 326 (2007) 1472-1475.

V. Nikiforov,Energy of matrices and norms of graphs, in: I. Gutman, X. Li (Eds.), GraphEnergies-Theory and Applications,Univ. Kragujevac, Kragujevac, 2016, pp. 5-48.


References

T. S. Shores,Linear Algebra and Matrix Analysis,Springer, New York, 2007.

M. P. Stanic, I. Gutman,On almost-equienergetic graphs,MATCH Commun, Math. Comput. Chem. 70 (2013) 681-688.

M. P. Stanic, I. Gutman,Towards a definition of almost-equienergetic graphs,J. Math. Chem. 52 (2014) 213-221.


THANK YOU FOR YOUR ATTENTION!

Ivan Gutman1,2, Boris Furtula1, Emir Zogic2, Edin Glogic2

1 Faculty of Science, University of Kragujevac, Kragujevac, Serbia

[email protected] , [email protected]

2 State University of Novi Pazar, Novi Pazar, Serbia

[email protected] , [email protected]


Documents

Resolvent Energy of Graphs - MISANU · Resolvent Energy of Graphs I.Gutman1;2, B.Furtula1, E.Zogi c2, E.Glogi c2 1 Faculty of Science, University of Kragujevac, Kragujevac, Serbia