SOME TOPICS IN SPECTRA OF...
Transcript of SOME TOPICS IN SPECTRA OF...
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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
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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
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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
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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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Date:
Place:
Narayan Swamy
(Research scholar)
Dr. H.B.Walikar
(Research Guide)