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Level Polynomials of Rooted Trees
Bünyamin Şahin *
Posted Date: 12 December 2023
doi: 10.20944/preprints202312.0902.v1
Keywords: Level index; Level polynomial; Triangular Numbers; Subdivision of Stars; Dendrimers
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Article
Level Polynomials of Rooted Trees
Bünyamin Şahin
Department of Mathematics, Faculty of Science, Selçuk University, Konya, TURKEY;
bunyamin.sahin@selcuk.edu.tr
Abstract: Level index was introduced in 2017 for rooted trees which is a component of Gini index.
In the origin, Gini index is a tool for economical investigations but Balaji and Mahmoud defined the
graph theoretical applications of this index for statistical analysis of graphs. Level index is an
important component of Gini index. In this paper we define a new graph polynomial which is called
level polynomial and calculate the level polynomial of some classes of trees. We obtain some
interesting relations between the level polynomials and some integer sequences.
Keywords: level index; level polynomial; triangular numbers; subdivision of stars; dendrimers
1. Introduction
The Gini index was defined by Corrado Gini in 1912 [1]. It shows the income inequality of social
groups and is used by The World Bank for the economical investigations. The graph theoretical
application of Gini index introduced by Balaji and Mahmoud in 2017 [2] for rooted trees. They
introduced two distance based topological index Gini index and level index. Moreover, degree based
Gini index was defined by Domicolo and Mahmoud in 2019 [3].
The first distance based topological index was introduced by Wiener in 1947 [4]. Wiener showed
that there is a correlation between the physico chemical properties of molecules and distances
between the atoms. Haruo Hosoya defined a distance counting polynomial in 1988 [5] which is
called Hosoya polynomial in the literature. The first derivative of Hosoya polynomial gives Wiener
index and second derive gives the Wiener polarity index. Derivatives of Hosoya polynomial were
used as molecular descriptors by Konstantinova and Diudea [6], Estrada et al. [7]. Moreover vertex-
weighted Wiener polynomials were studied by Doslic [8].
The level concept was used in the papers [9] and [10] for rooted trees. In [9] Flajolet and
Prodinger obtained a number sequence and investigated properties of this sequence. Statistical
analysiss of level numbers were studied by Balaji and Mahmoud [2].
In this paper we define a new distance based graph plynomial which is called “Level
Polynomial”. The first derivative of level polynomial gives the level index of graphs. Moreover, we
compute the level polynomial and level index of triangular numbers, caterpillar graphs, subdivisons
of stars and regular dendrimer graphs. We obtain some interesting relations between the coefficients
of level polynomials of graphs and some integer sequences.
2. Preliminaries
We use only simple, connected and undirected graphs. The degree of a vertex is denoted by
. A vertex with degree one is named a leaf. The notation is used to show the distance
between two any vertices and in a graph.
In a graph , the number of vertices is called order. The path and star graphs with
vertices are denoted by and , respectively.
Definition 2.1. The total distance from a vertex to other vertices is presented by the
following phrase
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from any ideas, methods, instructions, or products referred to in the content.
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© 2023 by the author(s). Distributed under a Creative Commons CC BY license.
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Definition 2.2. The Wiener index for a graph is defined by the following equation [4]
Definition 2.3. The Hosoya (Wiener) polynomial of a graph is denoted by and it is
computed by the following equation where denotes the vertex pairs having distance [5]
Theorem 2.4. The Hosoya polynomials of paths, stars, cycles with even order and odd order are
presented as follows
Theorem 2.5. The Wiener indices of paths, stars and cycles are presented in the following
equations
Definition 2.6. The Wiener index of a graph is also computed by the following equation [5]
3. Level Index and Dendrimer Graphs
In a rooted tree, a vertex determined as a root or central vertex. The distance from the central
vertex is denoted by [2]. This distance (measured with edges) is called by level. The distance
from the root to a vertex with the highest level is called height of the tree [2].
Balaji and Mahmoud introduced two distance based topological indices for rooted trees [2]. The
first one is called level index and level index of a tree is denoted by . Level index of a tree is
computed by the following equation
such that and showing the vertices at distances and from the central vertex of the
tree .
In order to exemplify the level index, we use the example given in the paper [2].
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Figure 1. The tree which is used in the following example.
Example 3.1. The level index of is computed by
Now we can describe a level counting polynomial which is called level polynomial of the graphs.
Definition 3.2. The level polynomial of a rooted tree is given by
where shows the number of vertex pairs having level difference . It is understood that level
index of a graph equals to
Lemma 3.3. For a given dendrimer graph (depicted in Figure 2) with central vertex , the
following properties are hold [11]
The order of is
,
contains branches,
Every branch of contains
vertices,
Every branch of contains leaves,
Every branch of contains
non-leaf vertices,
There are vertices at distance from .
Figure 2. Dendrimers and .
4. Main Results
In this section we obtain the main results of the paper. We obtain the level polynomial of some
classes of graphs. If denotes the number of vertices on level , we can show the level
polynomials of rooted trees as in the following theorem. Even though there exists one vertex at first
level () in a rooted tree, the definition of level polynomial can be extended to other graphs
and can take different values in the future.
Theorem 4.1. The level polynomial of a rooted tree is obtained by the following equation
such that the number of vertices on level is denoted by
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Proof. If the height of a rooted tree is showed by , the exponents of changes from to
. Since the level polynomial of a rooted tree can be presented as
The main problem is finding the coefficients of the level polynomial of . Since a level has to be
greater than 1, there is no constant term in the level polynomial.
The coefficient of is , because the vertex pairs which have level difference are located
on level 0 and level . Similarly The coefficient of is, because the vertex pairs
which have level difference are located on levels and levels .
By this way we obtain the coefficient of as , because we want to
obtain the number of vertices which have level difference 2.
Finally the coefficient of is . Because the vertices which have level
difference 1 are located at consecutive levels. It means that the level polynomial of a rooted tree is
presented as follows
Remark 4.2. The level index of a rooted tree equals to following equation by Defnition 3.2
We can find the level polynomials of trees which represent the triangular numbers as in the
following figure.
Let be a tree which has vertices on the level (depicted in Figure 3 for ). It
means that there is a central vetrex, two vertices on first level, three vertices on the second level and
vertices on -th level (triangular numbers). The sum of coefficient of level polynomial of
gives a new application of the integer sequence A000914 from OEIS.
Figure 3. The tree for .
Theorem 4.3. Assume that is defined above. Then its level polynomial is defined as follows
Proof. For a given sequence , the level partitions are defined as in the following
phrases, (Level )
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(Level )
(Level )
(Level )
(Level ).
By these phrases for a given level , the coefficients are ordered. Then the level
polynomial of is presented as in the following equation
Let be as in the previous theorem. Now it is denoted the sum of coefficients of level
polynomials by such that
Theorem 4.4. For a positive integer , the number of is computed as in the following
equation
Proof. In order to find the level number of a positive integer , we use Therorem 4.3 in
obtaining the sum of coefficients of Level polynomials of the tree .
Now we obtain the initial terms of the sequence of . For a positive integer , there is a tree
which has levels and there are vertices on level (. By this way initial
terms are obtained as
Theorem 4.5. The level index of the tree is defined in the following equation
Proof. In order to find the level index of a positive integer , we use Remark 4.2 in obtaining
the sum of coefficients of Level polynomials of the tree .
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Since the level polynomial of is given in the Theorem 4.2
If we obtain the initial terms of the sequence which is obtained in the Theorem 4.5
This sequence is appeared in the OEIS with reference number A067056 for level index of greater
than 1.
Theorem 4.6. Let be a tree with level . Assume that is a tree which is obtained from
by attaching a new vertex to -th level of . Then the difference between the level polynomials
of and is
Proof. Assume that a vertex is attached the -th level of . Then difference between the
level polynomials of and is
with the open form. We can write this equaion by
By the last equation, we can compute the difference of level indices of and .
Theorem 4.7. Let be a rooted tree. Then the level polynomial of equals to Hosoya
polynomial if and only if .
Proof. Since a path has one vertex at each level, there exists one vertex for each
distance from . Then the level polynomais of equals to Hosoya polynomial of as in the
following equation
Since the polynomials equal, we obtain that
Now we assume that . It means that and there are at least two vertices at a
level. Let such a level be -th level and two vertices and be two vertices at this level.
Therefore, and are at the same level and the difference of level equals to zero but the distance
between and is two. For the vertices which are located at the levels greater than , distances
from equal to level difference plus two. Then the Hosoya polynomial of is greater than Level
polynomial of as in the following equation.
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If the number of vertices which are the same level increases, then the difference
also increases.
Theorem 4.8. The level polynomial of a star of order equals to following equation
Proof. The star graph is consisted of a root and leaves at distance one from the root.
Then we obtain the level polynomial and level index of as follows
Let be a caterpillar graph which is defined in .
is obtained from a path by attaching leaves to vertices of paths as the
leaves located at consecutive level. It means that is root, at level for there are
vertices, and at the level there are leaves. To easify the notation we can write
instead of .
Theorem 4.9. The level polynomial of a caterpillar graph equals to
Proof. The distance can be obtained between the root and
Then we can write the level polynomial of caterpillar graph
We compute the level index of caterpillar as in the following equation
The last equation equals to level index of caterpillar which is obtained by direct calculation in
the paper .
Corollary 4.10. If it is taken , the level polynomial and level index of a
caterpillar graph are given in the following equations
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Corollary 4.11. If it is taken as , the following equations are obtained
Corollary 4.12. If it is taken , the level polynomial and level index of a
caterpillar graph are given in the following equations
Theorem 4.13. The level polynomial of tree of order is computed by the following equation
Proof. The tree is consisted a central vertex and paths which are attached to
(see Figure 4). It means that .
Figure 4. The tree (Subdivisions of star graph).
The level index of can be computed from the first derivative of .
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By this equation we obtain the level index of as in the following equation.
Theorem 4.14. The level polynomial of dendrimer (depicted in Figure 2) graph of order
is computed by the following equation
Proof. We use Theorem 4.1 for the level polynomial of dendrimer graph
If the previous equation is written in a closed form, we obtain that
To compute the level index of dendrimer graph , we can take the first derivative of level
polynom of .
This equation can be restated as follows.
The first term and second term of are showed by the following equation.
The third term of the is restated by the following equations
(for )
(for )
(for )
(for )
(for )
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We can take for easy writing of the equations. By this way we obtain the equation
as in the short equation
The third term of the can be written as follows
such that
It follows from the fact that the coefficients of are symmetric around ,
and from the fact that the coefficients of odd powers sum to
squares and of even powers to twice the triangular numbers.
The first ten coefficients of are 1, 2, 4, 6, 9, 12, 16, 20, 25, 30 which are the first terms of an
interesting integer sequence which is appeared in OEIS by reference number A002620.
Finally the level index of the dendrimer graph equals to
Conclusion
In this paper, we define a new graph polynomial which is based on the level index of rooted
trees. The level index was defined by Balaji and Mahmoud for statistical analysis of graphs. It is used
to measure balancing of rooted trees.
We show that level index can be calculated by level polynomials of graphs. We obtain the level
polynomial and level index of trees which represent the triangular numbers. The sum of coefficients
of level polynomials and level index of triangular numbers correspond some integer sequences
appeared in OEIS [12]. Moreover, we compute the level polynomial and level index of caterpillar
graphs, subdivision of star graphs and dendrimer graphs.
It is clear that level polynomial concept can be applied to rooted trees which represent the square
numbers, pentagonal numbers, hexagonal numbers and others. We know that Pascal triangle can be
represented by a perfect binay tree. Then, level polynomials can be applied to many integer objects.
References
1. Gini, C.; Veriabilità e Mutabilità. Cuppini, 1912, Bologna.
2. Balaji, H.; Mahmoud, H.; The Gini Index of Random Trees with Applications to Caterpillars, J.
Appl. Prob. 54 (2017), 701-709.
3. Domicolo, C.; Mahmoud, H.M.; Degree Based Gini Index for Graphs, Probability in the Engineering
and Informational Sciences 34 (2) (2019): 1-15.
4. Wiener, A.H.; Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69 (1947), 17-
20.
5. Hosoya, H.; On some counting polynomials in chemistry, Discrete Applied Mathematics 19 (1988), 239-
257.
6. Konstantinova, E.V.; Diudea, M.V., The Wiener Polynomial Derivatives and Other Topological
Indices in Chemical Research, Croatica Chemica Acta 73 (2) (2000), 383-403.
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7. Estrada, E.; Ovidiu, I.; Gutman, I; Gutierrez, A.; Rodriguez, L.; Extended Wiener indices. A new set
of descriptors for quantitative structure-property studies, New J. Chem (1998), 819-822.
8. Doslic, T.; The vertex-weighted Wiener polynomials for composite graphs, Ars Mathematica
Contemporanea 1 (2008), 66-80.
9. Flajolet, P.; Prodinger, H.; Level number sequences for trees, Discrete Mathematics 65 (1987), 149-156.
10. Tangora, M.C.; Level number sequences for trees and the lambda algebra, Europen J. Combinatorics
12 (1991), 433-443.
11. Şahin, B.; Şener, Ü.G., Total domination type invariants of regular dendrimer. Celal Bayar
University Journal of Science 16 (2) (2020), 225-228.
12. Sloane, N. J. and Ploufe, S.; The Encyclopedia of Integer Sequences, Academic Press, 1995, http://oeis.org.
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those
of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s)
disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or
products referred to in the content.
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 12 December 2023 doi:10.20944/preprints202312.0902.v1
