1

Synopsis of the Ph. D. Thesis Entitled

SOME TOPICS IN

SPECTRA OF GRAPHS

Submitted to the

Karnatak University, Dharwad

For the award of the degree of

DOCTOR OF PHILOSOPHY

IN

MATHEMATICS

By

Narayan Swamy

Research Guide

Dr. H. B. Walikar M.A., Ph.D.

Professor & Head

Department of Computer science,

Karnatak University

Dharwad, India

2

Synopsis of the Ph. D. thesis entitled “Some Topics in Spectra of Graphs”

submitted to Karnatak University, Dharwad by Narayan Swamy

This work has been carried out by Narayan Swamy under the Guidance of

Dr. H. B. Walikar .

Spectral graph theory has a long history. In the early days, matrix theory

and linear algebra were used to analyze adjacency matrices of graphs. Algebraic

methods are especially effective in treating graphs which are regular and

symmetric. Sometimes, certain eigenvalues have been referred to as the

“algebraic connectivity” of a graph . There is a large literature on algebraic

aspects of spectral graph theory, well documented in several surveys and books,

such as Biggs [3], Cvetkovi´c,Doob and Sachs [16], and Seidel .The eigenvalues

are closely related to almost all major invariants of a graph, linking one extremal

property to another.

Spectra of graphs have appeared frequently in the mathematical literature since

the fundamental papers of L. M. Lihtenbaum (1956) [35] and of L. Collatz and

U. Sinogowitz (1957) [10]. Even earlier starting from the work of E. Huckel

(1931) [33], theoretical chemists were interested in graph spectra to study the

stability of certain non-saturated hydrocarbons as well as other chemically

relevant facts.

Applications of graph spectra in combinatorics and graph theory

and also in other branches of sciences stimulate the investigation of relations

between spectral and structural properties of a graph. From the spectrum one

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can obtain the number of vertices, edges, loops, triangles, spanning trees [5, 31],

one can obtain the bounds for chromatic number and diameter [6, 7, 30]. The

Sachs theorem gives us the coefficients of the characteristic polynomials, which

are expressed in terms of graph structure [40]. There are lot of results giving

the relation between the spectrum of a graph obtained by some operations on

other graphs and spectra of graphs on which the operations done [6, 8, 19, 28,

29]. Some relations between the automorphism group of a graph and its

spectrum are given in [1].

Since the spectrum of a graph is unchanged by relabeling of its vertices,

early investigators were hoping that the spectrum would uniquely determine the

structure of the graph. But this was soon found to be false.

Since the spectrum of a graph is unchanged by relabeling of its vertices, early

investigators were hoping that the spectrum would uniquely determine the

structure of the graph. But this was soon found to be false. The smallest pair of

cospectral graphs (i.e. the graphs with same spectra) found to be star K1,4 and the

graph C4 U K1. Smallest connected pair of cospectral graphs has six vertices [5].

It is known that almost all trees have a cospectral mate even under some

additional restrictions [18]. Thus, in general, graph is not characterized by its

spectrum. But some spectral characterizations are possible [6, 15, 31, 41, 42].

Eigenvalues can be used for classifications and ordering of graphs. Strongly

regular graphs are classified by their parameters or by their distinct eigenvalues,

since the parameters are uniquely determine the eigenvalues and vice versa.

Similarly the block designs are classified by their parameters and spectrum.

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The construction of graphs with certain remarkable properties can be

performed in certain cases using graph spectra [10]. W.T. Tutte [43] proved that

the characteristic polynomial of a graph G can be uniquely reconstructed from

the collection of vertex deleted subgraphs of G.

Eigenvalues of certain graphs occurring in coding theory can be

expressed as Krawtchouk polynomials of some graph parameters [11].

Characteristic polynomials of cycles and of paths are infact the Chebyshev’s

polynomials of the first kind and of the second kind respectively. Hermite and

generalized Laguerre polynomials have an interpretation as the so-called acyclic

polynomials, which are closely related to characteristic polynomials of a

complete graph and complete bipartite graph [20].

In this work we defined new definition golden graphs, pure golden graphs

and almost pure golden graphs and got new results. We have constructed a class

of graphs which are golden graphs. It consists of 5 chapters.

Chapter 1: Introduction

Chapter 1 traces in brief the spectra of graphs and its development. Also it

recollects the definitions and terminologies used in the thesis.

Chapter 2: Golden Graphs-I

The Golden ratio has fascinated western philosophers, mathematicians, artists,

scientists and almost all intellectuals in all walks of life for atleast 2,400 years.

Johannes Kepler , a famous physicist and astronomer in sixteenth century,

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exclaimed ,”Geometry has two great treasures: one the Theorem of Pythagoras

and the other is the division of a line into extreme and mean ratio , the first one

has the value of the gold and second one is a precious jewel. Many western

intellectuals have come across the golden ratio their chosen field of

specialization and they used different names such as golden section, extreme and

mean ratio, medial section, divine proposition , divine section , golden number,

golden proposition and mean of Phidias. In Graph theory, graph theorist traced

the path of golden ratio in graphs with reference to various properties of graphs,

which we mention in chronological order[45].

The first one to deal with GR is W.T.Tutte(1970) in connection with

chromatic polynomials of graphs published in journal of combinatorial theory.

Michel O Albertson (1973) deals with boundary values in chromatic graph

theory.

Pavel Chebotarev (2008) come across GR with spanning forest in graphs.

Saeid Alikhani and Yee-hock (2009) deal with GR negatively with chromatic

zeros.

We are giving an account of GR in graphs which we came across while studying

the spectral properties of graphs. While studying the spectra of 4P , a path on 4

vertices , we find that its eigenvalues are 2

51,

2

51,

2

51,

2

51 which

are nothing but golden ratio (Divine ratio). Interestingly, we asked the question

which graphs have eigenvalues as golden ratio. In this chapter we have proved

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logically that, there are infinite class nP , a path on n vertices which are golden

ratio as eigen ratio and a path 4P is the only pure golden tree.

We defined pure golden graph and Golden graph as follows,

Definition.2.1: A graph G is said to be Pure golden graph, if all the Eigen

values of G are Golden ratios (i.e. 2

51,

2

51,

2

51 and

2

51)

Definition.2.2: A graph G is said to be golden graph, if at least one of the

eigenvalues of G are Golden ratios.

Result 2.1: A graph G is a pure golden tree if and only if 4PG , a path on

four nodes.

Result 2.2: An acyclic graph G is pure golden if and only if every

component of G is 4P .

Result 2.2: nP is golden graph if and only if 15 kn .

Examples 2.3: For the following graphs eigenvalues are

1) For 9P spectra is 1.902, 1.6180, 1.1761, 0.6180, 0, -1.902, -1.1761, -

0.6180, -1.6180

2) For 14P spectra is -1.9563, -1.8271, -1.6180, -1.3383, -1, -0.6180, -

0.2091, 0.2091, 0.6180, 1, 1.3383, 1.6180, 1.8271, 1.956.

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Chapter 3: Golden Graphs-II

While studying the spectra of 5C , a cycle on 5 nodes ,we find that its Eigen

values are 2,2

51,

2

51, and with multiplicities as 1,2,2 respectively

which are nothing but Golden ratio (Divine ratio).Interestingly ,we asked the

question which graphs have Eigen values as Golden ratio and also which graphs

has Eigen values as ,2

51,

2

51,where

2

51,

2

51 which we

defined as almost pure golden graphs .In this paper we have proved logically

that, there are infinite class of nC , a cycle on n nodes which have GR as eigen

value and cycle 5C is the only almost pure golden graph.

We defined almost pure golden graph as follows,

Definition 3.1: A graph G is said to be almost Pure golden graph, if all the

eigenvalues of G are ,2

511

,

2

512

, where

2

51,

2

51 .

Result 3.1: A connected graph G is almost pure golden graph if and only if

5CG ,a cycle on five vertices.

Result 3.2: An graph G is almost pure golden if and only if every component of

G is 5C .

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Result 3.2: nC is golden graph if and only if kn 5 .

Example 3.3: For the following graphs eigenvalues are

1) For 10C spectra is -2, -1.6180, 0.6180, 0.6180, 1.6180, 1.6180, 2, -

0.6180, -0.6180

2) For 15C spectra is -1.9563, -1.6180, -1.6180, -1, -1, -0.2091, -0.2091,

0.6180, 0.6180, 1.3383, 1.3383, 1.8271, 1.8271, 1, 2.

Chapter 4: Construction of golden graphs-I

In this chapter we have constructed golden graphs. First we have proved

logically that 4PG , 5CG and tree with diameter 6 as golden graphs. And

using the graph 4PG , we have constructed golden graphs by taking k copies of

4PG and each copy of 4PG is attached to isolated vertex. And also

constructed golden graphs using 5CG and tree with diameter 6 with the same

construction. In the end of this chapter, we have constructed golden graphs

using 5C , 15 kC and 14 kP .

Result 4.1: Let G be any graph, then 4PG is golden graph.

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Example 4.2:

FIGURE 4.1

Spectra of the graph [figure 4.2] is -1.8255, -1.6180, -1, -1, -1.4179, 0.6180,

5.2434

Result 4.3: Let G be any graph, then 5CG is golden graph.

Example 4.4:

FIGURE 4.2

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The eigen values of the graph [as in figure 4.2] is -1.8730, -1.6180, -1.6180, -1, -

1, 0.6180, 0.6180, 5.8730.

Definition 4.4: Let G be a graph obtained by taking k copies 15 KC and

attaching each copy of 15 KC of vertex of degree 5 to an isolated vertex u as

shown in the figure 4.3.

FIGURE 4.3

Result 4.3: Let G be graph as shown in the figure 4.3, then G golden graph.

u

w

k copies

11

Example 4.4:

FIGURE 4.4

The eigenvalues of the graph [figure 4.4] are -1.9793, -1.6180, -16180, -1.6180,

-1.4694, -1.1303, 0.0808, 0.6180, 0.6180, 0.6180, 1.1057, 3.4817, and 3.9108.

Definition 4.5: Let G be a graph obtained by taking k copies 14 KP and

attaching each copy of 14 KP of vertex of degree 4 to an isolated vertex u as

shown in the figure 4.5

.

FIGURE 4.5

Result 4.6: Let G be a graph as in the figure 4.5, then G is golden graph.

.

. .

k copies

u

w

12

Example 4.7:

FIGURE 4.6

The spectra of the graph [figure 4.6] is 3.0437, 2.9354, -1.8241, -1.4728,

0.3285, -0.5482, 0.4626, -1.6180, -1.6180, 0.6180 and 0.6180.

Note: The graph shown below 1H is golden graph

FIGURE 4.7

The spectra of the graph 1H [figure 4.7] is 2.3028, 0.6180, 0, -1.3028 and -

1.6180

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Definition 4.8: Let G be a graph obtained by taking k copies of 1H (as in figure

4.7) and attaching each copy of 1H of vertex of degree 2 to an isolated vertex u

as shown in the figure 4.8.

FIGURE 4.8

Result 4.9: Let G be the graph as in the figure 4.11, then G is golden graph.

Example 4.10:

k copies

u

w

14

FIGURE 4.9

The spectra of the graph [figure 4.9] is -1.8976, -1.6180, -1.6180, -1.3028, -

0.4891, 0, 0.6180, 0.6180, 0.8493, 2.3028 and 2.5374.

Similarly we constructed golden graphs by using the graph 5C and also using

tree of diameter 6.

Chapter 5: Construction of golden graphs-II

In this chapter we have constructed some more golden graphs. First we have

proved logically that for which n (i.e. number of vertices) tree nZ (single headed

snake) and Prism nI are golden graphs. Next which Mobious ladder are golden

graphs and 15 KC k as golden graphs. And also for which value of kji ,, the tree

],,[ kjiT is golden graph .Similarly, for which vales of niii ,......,, 21 the tree

15

],.....,[ 21 niiiT is golden graphs. We have proved logically that the tree nA (double

headed snake) is not golden graph. We have proved the graph 21 GG , where 1G

is regular graph and 2G is prism as golden graphs and also 4PKn as golden

graphs. In the end We have constructed golden graphs using the prism, Mobious

ladder, trees ],,[ kjiT , ],.....,[ 21 niiiT , 15 KC k , 4PG and 5CG as we done in

the chapter 4.

Definition 5.1: Let nZG be the tree (Single headed snake) 2n vertices as

show in figure 5.1.

FIGURE 5.1

Result 5.2: The graph nZG with 2n vertices is not golden graph.

Definition.5.3: Let nAG be the tree (Double headed snake) with 4n vertices as

show in figure 5.2

1 2 3 n

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FIGURE 5.2

Result 5.4: Let nAG be graph with 4n vertices is a golden graph, if

15 kn .

Result 5.5 : Let nIG be a prism with vertices 3,2 kkn and is a golden

graph, if 10mod0n .

Definition 5.6 : Let G be graph obtained by taking k copies of a prism nI with

vertices 1,10 kkn and attaching each copy of prism nI of a vertex to an

isolated vertex u as shown below in figure 5.3

1 2 3 n

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FIGURE 5.3

Result 5.7: Let G be graph as shown in figure 5.3, then G is golden graph.

Result 5.8: Let 1G and 2G be regular graphs order 1n and 2n respectively. 2G

be golden prism , then 21 GG is golden graph.

Definition 5.9 : Let G be graph obtained by taking 4P and attaching to vertices

of degree 2 to every vertex of nK as shown below in figure 5.4

u

w

k copies

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FIGURE 5.4

Result 5.10: Let G be graph as in figure 5.9, then G is golden graph.

Result 5.11: Let nMG be Mobious ladder graph, G is golden if

4 for 4

5ofmultilpleisk

kn .

Result 5.12: The graph 15 KCG k is golden graph.

Similarly we constructed golden graphs by using the Mobious ladder, 4PG ,

5CG , 15 KCG k and also using trees ),,( kjiT , ),........,,( 21 niiiT .

1v

2v

3v

4v

.

.

.

1u

2u

3u

4nu

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References:

1. L. Babai, Automorphism group and category of cospectral graphs, Acta Math. Acad. Sci.

Hung., 31(1978), 295-306.

2. M. Behzad, G. Chartrand and L. Lisniak-Foster, Graphs and Digraphs, Wadsworth

Intrnational Group, Belmont, 1979.

3. N. Biggs, Algebraic Graph Theory, Cambridge University Press, Cambridge, 1974.

4. R. L. Brooks, C. A. B. Smith, A. H. Stone and W. T. Tutte, dissection of a renctangle into

squares, Duke Math. J., (1940), 312-340.

5. A. Cayley, A theorem on trees, Quart. J. Math., 23(1889), 376-378.

6. F. H. Clarke, A graph polynomial and its applications, Discrete Math., 3(1972), 305-313.

7. L. Collatz and U. Sinogowitz, Spektren endlicher Grafen, Abh. Math. Sem. Univ.

Hamburg, 21(1957), 63-77.

8. C. A. Coulson, On the calculation of the energy in unsaturated hydrocarbon molecules,

Proc. Cambridge Phil. Soc., 36(1940), 201-203.

9. C. A. Coulson, The calculation of resonance and localization energies in aromatic

molecules, J. Chem. Soc. (London), (1954), 3111-3115.

10. C. A. Coulson and G. S. Rushbrooke, Note on the method of molecular orbitals, Proc.

Cambridge Phil. Soc., 36(1940), 193-200.

11. D. M. Cvetkovic, Doctoral Thesis: Graphs and their spectra, Univ. Beograd Publ.

Elektrotehn. Fak. Ser. Mat. Fiz. No. 354 – No. 356 (1971), 1-50.

12. D. M. Cvetkovic, Chromatic number and the spectrum of a graph, Publ. Inst. Math.

(Beograd), 14 (28) (1972), 25-38.

13. D. M. Cvetkovic, Spectra of graphs formed by some unary operations, Publ. Inst. Math.

(Beograd), 19 (33) (1975), 37-41.

14. D. M. Cvetkovic, Some possible directions in further investigations of graph spectra, In :

Algebraic Methods in Graph Theory Vol.I (eds. L. Lovasz and V.T. Sos), North – Holland,

Amsterdam, 1981, pp.47-67.

15. D. M. Cvetkovic, M. Doob, I. Gutman and A. Torgasev, Recent Results in the Theory of

Graph Spectra, North-Holland,Amsterdam, 1988.

16. D. M. Cvetkovic, M. Doob and H. Sachs, Spectra of Graphs, Academic Press, New York,

1980.

17. I. Gutman and L. Pavlovic, The energy of some graphs with large number of edges, Bull.

Acad. Serbe Sci. Arts (Cl. Math. Natur.). 118(1999), 35.

18. D. M. Cvetkovic and J.H. vanLint, An elementary proof of Lloyd’s theorem, Proc. Kon.

Ned. Akad. v. Wet. A, 80 (1) (1977), 6-10.

20

19. E. A. Dinic, A. K. Kelmans and M.A. Zaitcev, Non-isomorphic trees with the same T-

polynomials, Inform. Process. Lett., 6 (1977), 73-76.

20. M. Doob, Graphs with a small number of distinct eigenvalues, Ann. New York Acad. Sci.,

175 (1970) No.1, 104-110.

21. M. Doob, On characterizing certain graphs with four eigenvalues by their spectra, Linear

Algebra and Appl., 3 (1970), 461-482.

22. I. Gutman, Bounds for total -electron energy of polymethines, Chem. Phys. Letters,

50(1977), 488 – 490.

23. I. Gutman, Acyclic systems with extremal Huckel -electron energy, Theoret. Chim. Acta

(Berl.), 45(1977), 79-87.

24. I. Gutman, The energy of a graph, Ber. Math. Stat. Sekt. Forschungszentrum Graz, 103

(1978),1-22.

25. I. Gutman, Some relations for the independence and matching polynomials and their

chemical applications, Bull. Acad. Serbe Sci. Arts (Cl. Math. Natur.), 105(1992), 39-49.

26. I. Gutman, Note on Turker’s approximate formula for total -electron energy of

benzeonoid hydrocarbons, ACH – Models in Chemistry, 133(4)(1996), 415-420.

27. I. Gutman, Hyperenergetic molecules graphs, J. Serb. Chem. Soc., 64(3)(1999),199-205.

28. F. Harary, Graph Theory, narosa Publishing House, New Delhi, 1996.

29. C. Hoede and X. Li, Clique polynomials and independet set polynomials of graphs,

Discrete Math., 125(1994), 219-228.

30. A.J. Hoffman, On eigenvalues and colorings of graphs, In: Graph Theory and Its

Applications (ed. B. Harris), Academic Press, New York-London, 1970, pp. 79-91.

31. A.J. Hoffman and D.K. Ray-Chaudhuri, On the line graph of a symmetric balanced

incomplete block design, Trans. Amer. Math. Soc., 116(1965), 238-252.

32. A. J. Hoffman and R. R. Singleton, On Moore graphs with diameters 2 and 3, IBM J. Res.

Develop., 4 (1960), 497-504.

33. E. Huckel, Quantentheoretische Beitrage zum Benzolproblem, Z. Phys, 70 (1931), 204-

286.

34. H. Hutschenreuther, Einfacher Beweis des Matrix - Gerust - Satzes der Netzwerktheorie,

Wiss. Z. TH Ilmenau, 13(1967), 403-404.

35. L. M. Lihtenbaum, Characteristic values of a simple graph, Trud 3-go Vses. matem. sezda,

tom 1 (1956), 135 - 136.

36. K.R. Parthasarathy, Basic Graph Theory, Tata McGrow - Hill Publ. Co. Ltd., New Delhi,

1994.

21

37. V. R. Rosenfeld and I. Gutman, A novel approach to graph polynomials, Match, 24(1989),

191-199.

38. D. H. Rouvray, The topological matrix in quantum chemistry, In : Chemical Applications

of Graph Theory (ed. A.T. Balaban), Academic Press, London, 1976, pp. 175-221.

39. D.E. Rutherford, Some continuant determinants arising in physics and chemistry, Proc.

Roy. Soc. (Edinburgh), A 62 (1947), 229-236.

40. H Sachs, Uber selbstkomplementare Graphen, Publ. Math. Debrecen, 9 (1962), 270-288.

41. A. J. Schwenk, Computing the characteristic polynomial of a graph, In : Graphs and

Combinatorics (eds. R. Bari and F. Harary), Lecture Notes in Mathematics, 406, Springer –

Verlag, Berlin – Heilderberg – New York, 1974, pp. 153 – 172.

42. A.J. Schwenk and R. J. Wilson, On the eigenvalues of a graph, In : Selected Topics in

Graph Theory (eds. L.W. Beineke and R. J. Wilson), Academic Press, New York, 1978, pp.

307-336.

43. W. T. Tutte, All the king's horses - A guide to reconstruction, In : Graph Theory and

Related Topics, Proc. Conf. held at Univ. of Waterloo, Waterloo, Ontario, 1977, (eds. J.A.

Bondy and U.S.R. Murty), Academic Press, New York, 1979, pp. 15-33.

44. D.B. West, Introduction to Graph Theory, Prentice Hall of India Pvt. Ltd., New Delhi,

1999.

45. H.B.Walikar, Sense in NonSense , A Phliomathematical Excursion. Penrose publication,

2011.

Date:

Place:

Narayan Swamy

(Research scholar)

Dr. H.B.Walikar

(Research Guide)