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Entropy bounds for dendrimers
Applied Mathematics and Computation 242 (2014) 462–472
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Entropy bounds for dendrimers Zengqiang Chen a, Matthias Dehmer b,c, Frank Emmert-Streib d, Yongtang Shi e,a,⇑ a
College of Computer and Control Engineering, Nankai University, Tianjin 300071, PR China Institute for Bioinformatics and Translational Research, UMIT, Hall in Tyrol, Austria Department of Computer Science, Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, 85577 Neubiberg, Germany d Computational Biology and Machine Learning Laboratory, Center for Cancer Research and Cell Biology, School of Medicine, Dentistry and Biomedical Sciences, Faculty of Medicine, Health and Life Sciences, Queen’s University Belfast, Belfast, UK e Center for Combinatorics and LPMC-TJKLC, Nankai University, Tianjin 300071, PR China b c
a r t i c l e
i n f o
a b s t r a c t Many graph invariants have been used for the construction of entropy-based measures to characterize the structure of complex networks. When considering Shannon entropy-based graph measures, there has been very little work to find their extremal values. A reason for this might be the fact that Shannon’s entropy represents a multivariate function and all probability values are not equal to zero when considering graph entropies. Dehmer and Kraus proved some extremal results for graph entropies which are based on information functionals and express some conjectures generated by numerical simulations to find extremal values of graph entropies. Dehmer and Kraus discussed the extremal values of entropies for dendrimers. In this paper, we continue to study the extremal values of graph entropy for dendrimers, which has most interesting applications in molecular structure networks, and also in the pharmaceutical and biomedical area. Among all dendrimers with n vertices, we obtain the extremal values of graph entropy based on different well-known information functionals. Numerical experiments verifies our results. Ó 2014 Elsevier Inc. All rights reserved.
Keywords: Information theory Shannon’s entropy Graph entropy Dendrimers Extremal values
1. Introduction Exploring graph measures defined on the graph topology has been a fruitful research topic for decades [1–6]. Based on this research, many interdisciplinary areas such as mathematical chemistry, systems biology and mathematical psychology have been influenced when exploring network-based systems quantitatively, see, e.g., [1,7–9]. There are also some results and new techniques to characterize actual networks of contacts, new insights into the problem of how cooperative behavior arises and survives have been provided [10–15]. This is also a new direction for our further research. Studies of the information content of graphs and networks have been initiated in the late fifties based on the seminal work due to Shannon [16]. The concept of graph entropy [3,17] introduced by Rashevsky [18] and Trucco [19] has been used to measure the structural complexity of graphs [1,20,21]. The entropy of a graph is an information-theoretic quantity that has been introduced by Mowshowitz [22] and he interpreted it as the structural information content of a graph [22–25]. Here the complexity of a graph [26] is based on the well-known Shannon’s entropy [27,3,16,22]. Note the Körner’s graph entropy [28] has been introduced from an information theory point of view and has not been used to characterize graphs ⇑ Corresponding author. E-mail addresses: [email protected] [email protected] (Y. Shi).
(Z.
http://dx.doi.org/10.1016/j.amc.2014.05.105 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
Chen),
[email protected]
(M.
Dehmer),
[email protected]
(F.
Emmert-Streib),
Z. Chen et al. / Applied Mathematics and Computation 242 (2014) 462–472
463
quantitatively. A survey on graph entropy measures has been published by Dehmer and Mowshowitz [17]. A statistical analysis of topological graph measures has been performed by Emmert-Streib and Dehmer [29]. In view of the vast of amount of existing graph entropy measures [1,3], there has been very little work to find their extremal values [30]. A reason for this might be the fact that Shannon’s entropy represents a multivariate function and all probability values are not equal to zero when considering graph entropies. To the best of our knowledge, Dehmer and Kraus [30] were the first who determined minimal values of graph entropies. Still this problem is intricate because there is a lack of analytical methods to tackle this particular problem. Other related work is due to Shi [31], who proved a lower bound of quantum decision tree complexity by using Shannon’s entropy. Dragomir and Goh [32] obtained several general upper bounds for Shannon’s entropy by using Jensen’s inequality [33]. In [30], Dehmer and Kraus proved some extremal results for graph entropies which are based on information functionals. In particular, they derived information inequalities for dendrimers and statements regarding the extremality of the graph entropy by using majorization theory [34] and several information functionals [17]. Furthermore, they expressed some conjectures generated by numerical simulations to find extremal values of the mentioned graph entropies. The main contribution of the paper is to prove extremal results for dendrimers. In order to achieve our mathematical results, we employ graph entropies which are based on information functionals (see Section 2). The reason why we study dendrimers is that they have been proven useful in structural chemistry and in the pharmaceutical and biomedical area. For details on the theory of dendrimers, see [35–37]. 2. Preliminaries In the whole paper, ‘‘log’’ denotes the logarithm based on 2. In the following, we introduce entropy measures studied in this paper and state some preliminaries [3,38]. All measures examined in this paper are based on Shannon’s entropy. Definition 1. Let p ¼ ðp1 ; p2 ; . . . ; pn Þ be a probability vector, namely, 0 6 pi 6 1 and p is defined as
Pn
i¼1 pi
¼ 1. The Shannon’s entropy of
n X IðpÞ ¼ pi log pi : i¼1
To define information-theoretic graph measures, we often consider a tuple ðk1 ; k2 ; . . . ; kn Þ of non-negative integers ki 2 N [3]. This tuple forms a probability distribution p ¼ ðp1 ; p2 ; . . . ; pn Þ, where
ki pi ¼ Pn
j¼1 kj
i ¼ 1; 2; . . . ; n:
Therefore, the entropy of tuple ðk1 ; k2 ; . . . ; kn Þ is given by n n X X Iðk1 ; k2 ; . . . ; kn Þ ¼ pi log pi ¼ log ki i¼1
!
i¼1
n X i¼1
ki Pn
j¼1 kj
log ki :
ð1Þ
In the literature, there are various ways to obtain the tuple ðk1 ; k2 ; . . . ; kn Þ, like the so-called magnitude-based information measures introduced by Bonchev and Trinajstic´ [39], or partition-independent graph entropies, introduced by Dehmer [3,40], which are based on information functionals. We are now ready to define the entropy of a graph due to Dehmer [3] by using information functionals. Definition 2. Let G ¼ ðV; EÞ be a undirected connected graph. For a vertex
f ðv i Þ
pðv i Þ :¼ PjVj
v
j¼1 f ð j Þ
v i 2 V, we define
;
where f represents an arbitrary information functional. PjVj Observe that i¼1 pðv i Þ ¼ 1. Hence, we can interpret the quantities pðv i Þ as vertex probabilities. Now we immediately obtain one definition of graph entropy of a graph G. Definition 3. Let G ¼ ðV; EÞ be a undirected connected graph and f be an arbitrary information functional. The entropy of G is defined as jVj X f ðv i Þ f ðv i Þ If ðGÞ ¼ log PjVj PjVj i¼1 j¼1 f ðv j Þ j¼1 f ðv j Þ
!
¼ log
jVj X i¼1
!
f ðv i Þ
jVj X i¼1
f ðv i Þ log f ðv i Þ: PjVj j¼1 f ðv j Þ
ð2Þ
There is a large number of information functionals which can be used, see [3,17,41,40]. 3. Extremal values for graph entropies of dendrimers A dendrimer is a tree with 2 additional parameters, the progressive degree t and the radius r (as an example, see Fig. 1). Every internal node of the tree has degree t þ 1. As in every tree, a dendrimer has one (monocentric dendrimer) or two
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Fig. 1. The monocentric homogeneous dendrimer with t ¼ 3 and r ¼ 3.
(dicentric dendrimer) central nodes, the radius r denotes the (largest) distance from an external node to the (closer) center. If all external nodes are at distance r from the center, then the dendrimer is called homogeneous. Internal nodes different from the central nodes are called branching nodes and are said to be on the ith orbit if their distance to the (nearer) center is r. Every branching vertex has one incoming edge as well as t outgoing edges. Observe that
n¼1þ
ðt þ 1Þðt r 1Þ : t1
ð3Þ
From (3), we can deduce that
r¼
log ntnþ2 tþ1 : log t
ð4Þ
We will consider the extremal values for graph entropies based on different information functionals among all dendrimers. 3.1. Extremal values for I f r ðDÞ The eccentricity of v is rðv Þ ¼ maxu2V dðu; v Þ, where dðu; v Þ is the distance between vertices u and v. Let G ¼ ðV; EÞ. For a vertex v i 2 V, we define f as
f ðv i Þ :¼ ci rðv i Þ; where ci > 0 for 1 6 i 6 n. The entropy based on f, denoted by I f r ðGÞ, is defined as follows:
Ifr ðGÞ ¼ log
n X
!
ci rðv i Þ
i¼1
n X i¼1
ci rðv i Þ Pn logðci rðv i ÞÞ: j¼1 c j rðv j Þ
By performing numerical experiments, Dehmer and Kraus [30] obtained the following conjecture. Unfortunately, several attempts to prove the statement with different methods failed. Conjecture 1. Let D be a dendrimer on n vertices. For all sequences c0 P c1 P P cr , star graph ðr ¼ 1; t ¼ n 2Þ have maximal and path graph r ¼ bn1 2 c; t ¼ 1 have minimal entropy. As a special case, we consider ci ¼ cj for all i – j. By some elementary calculations, we have r X
r X
i¼1
i¼1
ri ¼ r þ
ðt þ 1Þt i1 ðr þ iÞ ¼ 2rn þ
2r n þ 1 : t1
Then we have
Ifr ðDÞ ¼ log
n X
!
ri
i¼1
n X ri log ri Pn i¼1
j¼1
rj
! r X 2r n þ 1 t1 i1 ¼ log 2rn þ r log r þ ðt þ 1Þ t ðr þ iÞ logðr þ iÞ : t1 2rnðt 1Þ þ 2r n þ 1 i¼1
ð5Þ
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As an example, in Fig. 2, we plot the values of I f r ðDÞ for n ¼ 104 and 1 6 t 6 n 2. As we have seen, the situation to determine the extremal values of I f r ðDÞ is intricate. Thus we can not conclude straightforwardly which graph attains the maximum value or the minimum value of I f r ðDÞ. Actually, Conjecture 1 does not hold in general and we conjecture that the star graph has minimal entropy in this case. Conjecture 2. Let D be a dendrimer on n vertices. For all sequences ci ¼ cj for all i – j, the star graph ðr ¼ 1; t ¼ n 2Þ has the minimal value of I f r ðDÞ. 3.2. Extremal values for I f k ðDÞ Let G ¼ ðV; EÞ be a connected graph with n vertices, m edges and degree sequence ðd1 ; d2 ; . . . ; dn Þ, where di ¼ dðv i Þ for k 1 6 i 6 n. By setting f ðv i Þ ¼ di in (2), we can obtain a new entropy based on degree powers, denoted by Ifk ðGÞ
I f k ðGÞ ¼ log
n X k di i¼1
!
1 Pn
n X k k di log di ;
k i¼1 di i¼1
which was first introduced in [42]. For k ¼ 1, we have
If1 ðGÞ ¼ log ð2mÞ
n 1 X di log di : 2m i¼1
Theorem 1. Let D be a dendrimer on n vertices. The star graph and path graph attain the minimal and maximal value of I f 1 ðDÞ, respectively. Proof. Let D be a dendrimer on n vertices with parameters t and r. Observe that all the vertices of D have degree t þ 1 or 1. Since the number of vertices of degree 1 in D is ðt þ 1Þt r1 , the number of vertices of degree t þ 1 in D is n ðt þ 1Þt r1 . Then by the definition of If1 , we have
n ðt þ 1Þtr1 ðt þ 1Þ logðt þ 1Þ If1 ðDÞ ¼ log ð2n 2Þ : 2n 2 From (3), we can deduce that
tr1 ¼
nt n þ 2 : tðt þ 1Þ
ð6Þ
By substituting (6), we have
If1 ðDÞ ¼ log ð2n 2Þ
n 2 ðt þ 1Þ logðt þ 1Þ : 2n 2 t
Fig. 2. The values of entropies I f r ðDÞ for n ¼ 104 .
ð7Þ
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Since for any t P 1,
0 t logðt þ 1Þ ðt þ 1Þ logðt þ 1Þ ¼ ln 2 > 0; t t2 we have ðtþ1Þ logðtþ1Þ is a monotone increasing function on t, which implies that If1 ðDÞ is a monotone decreasing function on t. t Therefore, the result of the theorem holds. h For general k, we have
Ifk ðDÞ ¼ log
nt n þ 2 þ ðn 2Þðt þ 1Þk kðn 2Þðt þ 1Þk logðt þ 1Þ : t nt n þ 2 þ ðn 2Þðt þ 1Þk
The extremal values of I f k ðDÞ are much complicated, and we can not easily say exactly which graph attains the maximum value or the minimum value of I f k ðDÞ. As an example, in Fig. 3, we plot the values of I f r ðDÞ for n ¼ 104 ; k ¼ 10 and 1 6 t 6 n 2. 3.3. Extremal values for IV ðDÞ and IE ðDÞ The following entropy was introduced by Emmert-Streib and Dehmer in [43,44], which was called the vertex entropy. Let G be a generalized tree with hight h. Denote by jVj and jV i j the total number of vertices and the number of vertices on the ith level, respectively. A probability distribution of G is assigned as follows:
pVi ¼
jV i j : jVj 1
ð8Þ
Then the entropy of a generalized tree G is defined by
IV ðHÞ :¼
h X pVi logðpVi Þ: i¼1
Similarly, the edge entropy was also introduced in [43,44]. Denote by jEj and jEi j the total number of edges and the number of edges on the ith level, respectively. A probability distribution of G is assigned as follows:
pEi ¼
jEi j : jEj 1
Then the entropy of a generalized tree G is defined by
IE ðHÞ :¼
h X pEi logðpEi Þ: i¼1
Fig. 3. The values of entropies I f k ðDÞ for n ¼ 104 and k ¼ 10.
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Theorem 2. Let D be a dendrimer on n vertices. The star graph attains the minimal value of IV ðDÞ, and the dendrimer with parameter t ¼ t 0 attains the maximal value of IV ðDÞ, where t ¼ t 0 2 ð1; n 2Þ is the integer which is closest to the root of the equation
n nt n þ 2 tðt þ 1Þ 2t ln ln ¼ 0: n1 tþ1 n1 tþ1 Proof. From the definition of dendrimer, we have
pVi ¼
jV i j ðt þ 1Þt i1 ¼ : n1 n1
ð9Þ
Therefore,
r X ðt þ 1Þt i1 ðt þ 1Þti1 tþ1 ¼ þ ði 1Þ log t log n1 n1 n1 n1 i¼1 i¼1 ! r r X tþ1 t þ 1 X i1 log t þ log t ði 1Þt i1 ¼ n1 n 1 i¼1 i¼1 r1 tþ1 t þ 1 tr 1 t þ 1 tðt 1Þ log log t ðr 1Þt r : ¼ n1 n1 t1 n1 t1
IV ðGÞ ¼
r X ðt þ 1Þti1
log
ð10Þ
By substituting Eqs. (4) and (6) into (10), we have
tþ1 t þ 1 ðn 1Þðt 1Þ log n1 n1 t1 0 0 1 1 ntnþ2 log tþ1 tþ1 t nt t þ 2 nt n þ 2 A 1 @ log t @ 1A n1 t 1 tðt þ 1Þ tþ1 log t tþ1 nt2 nt n þ 2 nt n þ 2 ¼ ðt þ 1Þ log : log log t þ n1 n1 n1 tðt þ 1Þ
IV ðGÞ ¼
ð11Þ
By some elementary calculations and simplifications, we have
@ IV ðGÞ @t
1 ln t n nt n þ 2 tþ1 2t ln ln þ ln 2 n 1 n 1 tðt þ 1Þ n1 tþ1 1 n nt n þ 2 tðt þ 1Þ 2t ¼ ln ; ln ln 2 n 1 tþ1 n1 tþ1
¼
and
@ 2 ðIV ðGÞÞ ð3n2 2nÞt3 þ ð2n2 þ 4n 4Þt 2 ð7n2 16n þ 8Þt þ 2ðn 1Þðn 2Þ ¼ : 2 @t ðn 1Þðt þ 1Þ2 ðnt n þ 2Þt ln 2 Observe that
@ 2 ðIV ðGÞÞ @t 2
< 0 for all n and t. Then IV ðGÞ is a concave function on t. From Eq. (10), we know that IV ðGÞ ¼ 0
when t ¼ n 2, i.e., IV ðSn Þ ¼ 0. So, the star graph attains the minimal value of IV ðDÞ. On the other hand, the maximal value of IV ðDÞ will be attained when
n nt n þ 2 tðt þ 1Þ 2t ln ln ¼ 0: n1 tþ1 n1 tþ1
@ðIV ðGÞÞ @t
¼ 0, i.e.,
Therefore, the dendrimer with t ¼ t0 attains the maximal value of IV ðDÞ, where t ¼ t0 2 ð1; n 2Þ is the integer which is closest to the root of the equation
n nt n þ 2 tðt þ 1Þ 2t ln ln ¼ 0: n1 tþ1 n1 tþ1
Fig. 4 shows the relation between the values of n and t 0 . From Fig. 4, we can see that the exact value of t 0 is around 0:37n. Observe that for trees, the values of the vertex entropy and the edge entropy are equal. Then we have the following theorem for the edge entropy. Theorem 3. Let D be a dendrimer on n vertices. The star graph attains the minimal value of IE ðDÞ, and the dendrimer with t ¼ t 0 attains the maximal value of IE ðDÞ, where t ¼ t 0 2 ð1; n 2Þ is the integer which is closest to the root of the equation
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n nt n þ 2 tðt þ 1Þ 2t ln ln ¼ 0: n1 tþ1 n1 tþ1 3.4. Extremal values for the generalized graph entropies In [45], Dehmer and Mowshowitz introduced a new class of measures (called here generalized measures) that derived from functions such as those defined by Rényi’s entropy [46] and Daròczy’s entropy [47], which is now called the generalized graph entropies. We state the definition as follows. Let G be a graph of order n and X be a set of elements of a graph. Let s be an equivalence relation defined on X. Denote by X i and k the equivalence class and the number of equivalence classes of the relation s. Then,
jX i j ; jXj jXj i¼1 a ! n X 1 jX i j 2 ðiiÞ Ia ðGÞ :¼ log ; a – 1; 1a jXj i¼1 Pn jXi j a 1 i¼1 jXj 3 ; a – 1: ðiiiÞ Ia ðGÞ :¼ 1a 2 1
ðiÞ I1 ðGÞ :¼
n X jX i j
1
According to Rashevsky [18], jX i j denotes the number of topologically equivalent vertices in the ith vertex orbit of G, where k is the number of different orbits. Vertices are considered as topologically equivalent if they belong to the same orbit of a jV i j graph G. Now we suppose X ¼ jVj 1, and then the probability of X i can be expressed as pVi ¼ jVj1 , which is the same as
Eq. (8). Therefore, the entropies can be expressed as follows:
ðiÞ I1 ðGÞ :¼
n X pVi 1 pVi ; i¼1
! n X V a 1 ; a – 1; log pi ðiiÞ Ia ðGÞ :¼ 1a i¼1 Pn V a p 1 ; a – 1: ðiiiÞ I3a ðGÞ :¼ i¼11ai 2 1 2
Theorem 4. Let D be a dendrimer on n vertices. (i) The star graph and path graph attain the minimal and maximal value of I1 ðDÞ, respectively. (ii) For a – 1, the star graph and path graph attain the minimal and maximal value of I2a ðDÞ, respectively. (iii) For a – 1, the star graph and path graph attain the minimal and maximal value of I3a ðDÞ, respectively.
Fig. 4. The values n and t0 in Theorem 2.
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Proof. Pn
(i) Since I1 ðGÞ :¼
I1 ðGÞ ¼ ¼
V i¼1 pi
1 pVi , from Eq. (6), we infer
X 2 n n n X ðt þ 1Þt i1 ðt þ 1Þti1 ðt þ 1Þt i1 X ðt þ 1Þt i1 ðt þ 1Þðtr 1Þðn tr 2Þ ¼ ¼ 1 n 1 n 1 n 1 n 1 ðn 1Þ2 ðt 1Þ i¼1 i¼1 i¼1 2ðn t 2Þ : ðn 1Þðt þ 1Þ
ð12Þ
Thus, I1 ðDÞ is monotone decreasing on t, which implies that the star graph and path graph attain the minimal and maximal value of I1 ðDÞ, respectively.Now we are ready to show (ii) and (iii). From Eq. (9), we can obtain that
pVi ¼
ðt þ 1Þti1 : n1
Then from Eq. (6), we have r r X ðt þ 1Þa X ðt þ 1Þa ðt ar1 Þ ðnt n þ 2Þa ðt þ 1Þa a ¼ ðpVi Þ ¼ tði1Þa ¼ : a a a ðn 1Þ i¼1 ðn 1Þa ðta 1Þ ðn 1Þ ðt 1Þ i¼1
ð13Þ
By some elementary calculations and simplifications, we get
@
P r
V a i¼1 ðpi Þ
¼
@t
aðnt n þ 2Þa1 ðn 2Þta1 n þ aðt þ 1Þa1 ðta1 þ 1Þ ðn 1Þa ðt a 1Þ
2
:
Observe that nt n þ 2 P t þ 1, then ðnt n þ 2Þa1 P ðt þ 1Þa1 for a > 1 and ðnt n þ 2Þa1 6 ðt þ 1Þa1 for 0 < a < 1. Therefore, we have
@
P r
V a i¼1 ðpi Þ
P
@t
aðt þ 1Þa1 ðn 2Þta1 n þ aðt þ 1Þa1 ðta1 þ 1Þ a
a
ðn 1Þ ðt 1Þ
2
¼
aðn 1Þðt þ 1Þa1 ðta1 1Þ ðn 1Þa ðta 1Þ
2
>0
for a > 1, and
@
P r
V a i¼1 ðpi Þ
@t for 0 < a < 1. Then
6 Pr
ðnt n þ 2Þa1 ðn 2Þt a1 n þ aðnt n þ 2Þa1 ðt a1 þ 1Þ
V a i¼1 ðpi Þ
ðn 1Þa ðt a 1Þ
2
¼
aðn 1Þðnt n þ 2Þa1 ðta1 1Þ ðn 1Þa ðt a 1Þ
2
<0
is monotone increasing on t when a > 1 and monotone decreasing on t when 0 < a < 1.
(i) By the definition of I2a ðDÞ, we get that I2a ðDÞ is monotone decreasing on t for 0 < a < 1 and a > 1. Therefore, the star graph and path graph attain the minimal and maximal value of I2a ðDÞ, respectively.
Fig. 5. The values of entropies I f r ðDÞ (red), I f 1 ðDÞ (blue), IV ðDÞ (cyan), I1 ðDÞ (green), I2a ðDÞ (black) and I3a ðDÞ (grey), for a ¼ 0:5; t ¼ 2 (left) and t ¼ 3 (right). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 6. The values of entropies I f r ðDÞ (red), I f 1 ðDÞ (blue), IV ðDÞ (cyan), I1 ðDÞ (green), I2a ðDÞ (black) and I3a ðDÞ (grey), for a ¼ 2; t ¼ 2 (left) and t ¼ 3 (right). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
(ii) By the definition of I3a ðDÞ, we get that I3a ðDÞ is monotone decreasing on t for 0 < a < 1 and a > 1. Therefore, the star graph and path graph attain the minimal and maximal value of I3a ðDÞ, respectively.The proof is thus complete. h 4. Numerical Results In Fig. 5 and 6, we calculate the values of entropies I f r ðDÞ (red), I f 1 ðDÞ (blue), IV ðDÞ (cyan), I1 ðDÞ (green), I2a ðDÞ (black) and I3a ðDÞ (grey), considered in Section 3 for dendrimers with t ¼ 2 and t ¼ 3. From Fig. 5 and 6, we can see the change tendency of the value of each entropy according to the value of n. As we have seen from Fig. 5 and 6, for a ¼ 0:5 or a ¼ 2; t ¼ 2 or t ¼ 3, the following inequalities hold
IV ðDÞ > Ifr ðDÞ > If1 ðDÞ > I3a ðDÞ > I2a ðDÞ > I1 ðDÞ: On the other hand, the value of I f r ðDÞ is much larger than the other entropies. The difference between I f 1 ðDÞ and IV ðDÞ is not quite large. In general, the values of I1 ðDÞ; I2a ðDÞ and I3a ðDÞ are similar. In Fig. 2 and 3, we plot the values of I f r ðDÞ and I f k ðDÞ for n ¼ 104 and 1 6 t 6 n 2, respectively. As we have seen, the situation to determine the extremal values of I f r ðDÞ and I f k ðDÞ is complicated, and we can not easily say exactly which graph attains the maximum value or the minimum value of I f r ðDÞ and I f k ðDÞ, respectively. Simultaneously, according to the values
Fig. 7. The values of entropies I f 1 ðDÞ (left), IV ðDÞ (right) for n ¼ 104 .
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Fig. 8. The values of entropies I1 ðDÞ (left), I2a ðDÞ (middle), I3a ðDÞ (right) for a ¼ 0:5 and n ¼ 104 .
Fig. 9. The values of entropies I1 ðDÞ (left), I2a ðDÞ (middle), I3a ðDÞ (right) for a ¼ 2 and n ¼ 104 .
of t, we plot the values of I f 1 ðDÞ and IV ðDÞ for n ¼ 104 , respectively, as shown in Fig. 7. We also plot the values of I1 ðDÞ; I2a ðDÞ and I3a ðDÞ for n ¼ 104 and a ¼ 0:5 or a ¼ 2, respectively, as shown in Fig. 8 and 9. From Fig. 7–9, we can see that except IV ðDÞ, the star graph (t ¼ n 2) and path graph (t ¼ 1) attain the minimal and maximal value of entropies, respectively, which confirms the results of Theorems 1–4. 5. Conclusion In this paper, we tackled the problem of finding extremal values of graph entropies for dendrimers by using information functionals. This problem is highly relevant for designing new applications related to molecular structure networks, and also in the pharmaceutical and biomedical area. Among all dendrimers with n vertices, we obtain the extremal values of graph entropy based on several theorems (see previous section). From Theorems 1–4, we obtain that among all dendrimers with n vertices, the star graph (t ¼ n 2) and path graph (t ¼ 1) attain the minimal and maximal value of entropies, respectively. An exception is the entropy I f r ðDÞ and IV ðDÞ. The problem of determining the extremal values of I f r ðDÞ is intricate. It is interesting that, among all dendrimers with n vertices, the star graph attains the minimal value of IV ðDÞ, and the dendrimer with t ¼ t 0 attains the maximal value of IE ðDÞ, where t ¼ t 0 2 ð1; n 2Þ is the integer which is closest to 0:37n. Numerical experiments verified our mathematical findings. In the future, we would like to continue proving extremal results by employing graph entropies. In particular, it is interesting to investigate the extremal values of entropies for some other well-known network classes. A breakthrough would be to find analytical techniques to determine minimal graph entropies analytically for general graphs. The reason why this problem is still challenging relates to the fact that known results from information theory and majorization seem to be not applicable. Acknowledgments Zengqiang Chen was supported by the National Science Foundation of China (No. 61174094) and the Natural Science Foundation of Tianjin (No. 14JCYBJC18700). Matthias Dehmer and Yongtang Shi were supported by the Austrian Science Funds for supporting this work (Project P26142). Yongtang Shi was also are supported by NSFC, PCSIRT, China Postdoctoral Science Foundation (2014M551015) and China Scholarship Council.
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Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Entropy bounds for dendrimers Zengqiang Chen a, Matthias Dehmer b,c, Frank Emmert-Streib d, Yongtang Shi e,a,⇑ a
College of Computer and Control Engineering, Nankai University, Tianjin 300071, PR China Institute for Bioinformatics and Translational Research, UMIT, Hall in Tyrol, Austria Department of Computer Science, Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, 85577 Neubiberg, Germany d Computational Biology and Machine Learning Laboratory, Center for Cancer Research and Cell Biology, School of Medicine, Dentistry and Biomedical Sciences, Faculty of Medicine, Health and Life Sciences, Queen’s University Belfast, Belfast, UK e Center for Combinatorics and LPMC-TJKLC, Nankai University, Tianjin 300071, PR China b c
a r t i c l e
i n f o
a b s t r a c t Many graph invariants have been used for the construction of entropy-based measures to characterize the structure of complex networks. When considering Shannon entropy-based graph measures, there has been very little work to find their extremal values. A reason for this might be the fact that Shannon’s entropy represents a multivariate function and all probability values are not equal to zero when considering graph entropies. Dehmer and Kraus proved some extremal results for graph entropies which are based on information functionals and express some conjectures generated by numerical simulations to find extremal values of graph entropies. Dehmer and Kraus discussed the extremal values of entropies for dendrimers. In this paper, we continue to study the extremal values of graph entropy for dendrimers, which has most interesting applications in molecular structure networks, and also in the pharmaceutical and biomedical area. Among all dendrimers with n vertices, we obtain the extremal values of graph entropy based on different well-known information functionals. Numerical experiments verifies our results. Ó 2014 Elsevier Inc. All rights reserved.
Keywords: Information theory Shannon’s entropy Graph entropy Dendrimers Extremal values
1. Introduction Exploring graph measures defined on the graph topology has been a fruitful research topic for decades [1–6]. Based on this research, many interdisciplinary areas such as mathematical chemistry, systems biology and mathematical psychology have been influenced when exploring network-based systems quantitatively, see, e.g., [1,7–9]. There are also some results and new techniques to characterize actual networks of contacts, new insights into the problem of how cooperative behavior arises and survives have been provided [10–15]. This is also a new direction for our further research. Studies of the information content of graphs and networks have been initiated in the late fifties based on the seminal work due to Shannon [16]. The concept of graph entropy [3,17] introduced by Rashevsky [18] and Trucco [19] has been used to measure the structural complexity of graphs [1,20,21]. The entropy of a graph is an information-theoretic quantity that has been introduced by Mowshowitz [22] and he interpreted it as the structural information content of a graph [22–25]. Here the complexity of a graph [26] is based on the well-known Shannon’s entropy [27,3,16,22]. Note the Körner’s graph entropy [28] has been introduced from an information theory point of view and has not been used to characterize graphs ⇑ Corresponding author. E-mail addresses: [email protected] [email protected] (Y. Shi).
(Z.
http://dx.doi.org/10.1016/j.amc.2014.05.105 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
Chen),
[email protected]
(M.
Dehmer),
[email protected]
(F.
Emmert-Streib),
Z. Chen et al. / Applied Mathematics and Computation 242 (2014) 462–472
463
quantitatively. A survey on graph entropy measures has been published by Dehmer and Mowshowitz [17]. A statistical analysis of topological graph measures has been performed by Emmert-Streib and Dehmer [29]. In view of the vast of amount of existing graph entropy measures [1,3], there has been very little work to find their extremal values [30]. A reason for this might be the fact that Shannon’s entropy represents a multivariate function and all probability values are not equal to zero when considering graph entropies. To the best of our knowledge, Dehmer and Kraus [30] were the first who determined minimal values of graph entropies. Still this problem is intricate because there is a lack of analytical methods to tackle this particular problem. Other related work is due to Shi [31], who proved a lower bound of quantum decision tree complexity by using Shannon’s entropy. Dragomir and Goh [32] obtained several general upper bounds for Shannon’s entropy by using Jensen’s inequality [33]. In [30], Dehmer and Kraus proved some extremal results for graph entropies which are based on information functionals. In particular, they derived information inequalities for dendrimers and statements regarding the extremality of the graph entropy by using majorization theory [34] and several information functionals [17]. Furthermore, they expressed some conjectures generated by numerical simulations to find extremal values of the mentioned graph entropies. The main contribution of the paper is to prove extremal results for dendrimers. In order to achieve our mathematical results, we employ graph entropies which are based on information functionals (see Section 2). The reason why we study dendrimers is that they have been proven useful in structural chemistry and in the pharmaceutical and biomedical area. For details on the theory of dendrimers, see [35–37]. 2. Preliminaries In the whole paper, ‘‘log’’ denotes the logarithm based on 2. In the following, we introduce entropy measures studied in this paper and state some preliminaries [3,38]. All measures examined in this paper are based on Shannon’s entropy. Definition 1. Let p ¼ ðp1 ; p2 ; . . . ; pn Þ be a probability vector, namely, 0 6 pi 6 1 and p is defined as
Pn
i¼1 pi
¼ 1. The Shannon’s entropy of
n X IðpÞ ¼ pi log pi : i¼1
To define information-theoretic graph measures, we often consider a tuple ðk1 ; k2 ; . . . ; kn Þ of non-negative integers ki 2 N [3]. This tuple forms a probability distribution p ¼ ðp1 ; p2 ; . . . ; pn Þ, where
ki pi ¼ Pn
j¼1 kj
i ¼ 1; 2; . . . ; n:
Therefore, the entropy of tuple ðk1 ; k2 ; . . . ; kn Þ is given by n n X X Iðk1 ; k2 ; . . . ; kn Þ ¼ pi log pi ¼ log ki i¼1
!
i¼1
n X i¼1
ki Pn
j¼1 kj
log ki :
ð1Þ
In the literature, there are various ways to obtain the tuple ðk1 ; k2 ; . . . ; kn Þ, like the so-called magnitude-based information measures introduced by Bonchev and Trinajstic´ [39], or partition-independent graph entropies, introduced by Dehmer [3,40], which are based on information functionals. We are now ready to define the entropy of a graph due to Dehmer [3] by using information functionals. Definition 2. Let G ¼ ðV; EÞ be a undirected connected graph. For a vertex
f ðv i Þ
pðv i Þ :¼ PjVj
v
j¼1 f ð j Þ
v i 2 V, we define
;
where f represents an arbitrary information functional. PjVj Observe that i¼1 pðv i Þ ¼ 1. Hence, we can interpret the quantities pðv i Þ as vertex probabilities. Now we immediately obtain one definition of graph entropy of a graph G. Definition 3. Let G ¼ ðV; EÞ be a undirected connected graph and f be an arbitrary information functional. The entropy of G is defined as jVj X f ðv i Þ f ðv i Þ If ðGÞ ¼ log PjVj PjVj i¼1 j¼1 f ðv j Þ j¼1 f ðv j Þ
!
¼ log
jVj X i¼1
!
f ðv i Þ
jVj X i¼1
f ðv i Þ log f ðv i Þ: PjVj j¼1 f ðv j Þ
ð2Þ
There is a large number of information functionals which can be used, see [3,17,41,40]. 3. Extremal values for graph entropies of dendrimers A dendrimer is a tree with 2 additional parameters, the progressive degree t and the radius r (as an example, see Fig. 1). Every internal node of the tree has degree t þ 1. As in every tree, a dendrimer has one (monocentric dendrimer) or two
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Fig. 1. The monocentric homogeneous dendrimer with t ¼ 3 and r ¼ 3.
(dicentric dendrimer) central nodes, the radius r denotes the (largest) distance from an external node to the (closer) center. If all external nodes are at distance r from the center, then the dendrimer is called homogeneous. Internal nodes different from the central nodes are called branching nodes and are said to be on the ith orbit if their distance to the (nearer) center is r. Every branching vertex has one incoming edge as well as t outgoing edges. Observe that
n¼1þ
ðt þ 1Þðt r 1Þ : t1
ð3Þ
From (3), we can deduce that
r¼
log ntnþ2 tþ1 : log t
ð4Þ
We will consider the extremal values for graph entropies based on different information functionals among all dendrimers. 3.1. Extremal values for I f r ðDÞ The eccentricity of v is rðv Þ ¼ maxu2V dðu; v Þ, where dðu; v Þ is the distance between vertices u and v. Let G ¼ ðV; EÞ. For a vertex v i 2 V, we define f as
f ðv i Þ :¼ ci rðv i Þ; where ci > 0 for 1 6 i 6 n. The entropy based on f, denoted by I f r ðGÞ, is defined as follows:
Ifr ðGÞ ¼ log
n X
!
ci rðv i Þ
i¼1
n X i¼1
ci rðv i Þ Pn logðci rðv i ÞÞ: j¼1 c j rðv j Þ
By performing numerical experiments, Dehmer and Kraus [30] obtained the following conjecture. Unfortunately, several attempts to prove the statement with different methods failed. Conjecture 1. Let D be a dendrimer on n vertices. For all sequences c0 P c1 P P cr , star graph ðr ¼ 1; t ¼ n 2Þ have maximal and path graph r ¼ bn1 2 c; t ¼ 1 have minimal entropy. As a special case, we consider ci ¼ cj for all i – j. By some elementary calculations, we have r X
r X
i¼1
i¼1
ri ¼ r þ
ðt þ 1Þt i1 ðr þ iÞ ¼ 2rn þ
2r n þ 1 : t1
Then we have
Ifr ðDÞ ¼ log
n X
!
ri
i¼1
n X ri log ri Pn i¼1
j¼1
rj
! r X 2r n þ 1 t1 i1 ¼ log 2rn þ r log r þ ðt þ 1Þ t ðr þ iÞ logðr þ iÞ : t1 2rnðt 1Þ þ 2r n þ 1 i¼1
ð5Þ
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As an example, in Fig. 2, we plot the values of I f r ðDÞ for n ¼ 104 and 1 6 t 6 n 2. As we have seen, the situation to determine the extremal values of I f r ðDÞ is intricate. Thus we can not conclude straightforwardly which graph attains the maximum value or the minimum value of I f r ðDÞ. Actually, Conjecture 1 does not hold in general and we conjecture that the star graph has minimal entropy in this case. Conjecture 2. Let D be a dendrimer on n vertices. For all sequences ci ¼ cj for all i – j, the star graph ðr ¼ 1; t ¼ n 2Þ has the minimal value of I f r ðDÞ. 3.2. Extremal values for I f k ðDÞ Let G ¼ ðV; EÞ be a connected graph with n vertices, m edges and degree sequence ðd1 ; d2 ; . . . ; dn Þ, where di ¼ dðv i Þ for k 1 6 i 6 n. By setting f ðv i Þ ¼ di in (2), we can obtain a new entropy based on degree powers, denoted by Ifk ðGÞ
I f k ðGÞ ¼ log
n X k di i¼1
!
1 Pn
n X k k di log di ;
k i¼1 di i¼1
which was first introduced in [42]. For k ¼ 1, we have
If1 ðGÞ ¼ log ð2mÞ
n 1 X di log di : 2m i¼1
Theorem 1. Let D be a dendrimer on n vertices. The star graph and path graph attain the minimal and maximal value of I f 1 ðDÞ, respectively. Proof. Let D be a dendrimer on n vertices with parameters t and r. Observe that all the vertices of D have degree t þ 1 or 1. Since the number of vertices of degree 1 in D is ðt þ 1Þt r1 , the number of vertices of degree t þ 1 in D is n ðt þ 1Þt r1 . Then by the definition of If1 , we have
n ðt þ 1Þtr1 ðt þ 1Þ logðt þ 1Þ If1 ðDÞ ¼ log ð2n 2Þ : 2n 2 From (3), we can deduce that
tr1 ¼
nt n þ 2 : tðt þ 1Þ
ð6Þ
By substituting (6), we have
If1 ðDÞ ¼ log ð2n 2Þ
n 2 ðt þ 1Þ logðt þ 1Þ : 2n 2 t
Fig. 2. The values of entropies I f r ðDÞ for n ¼ 104 .
ð7Þ
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Since for any t P 1,
0 t logðt þ 1Þ ðt þ 1Þ logðt þ 1Þ ¼ ln 2 > 0; t t2 we have ðtþ1Þ logðtþ1Þ is a monotone increasing function on t, which implies that If1 ðDÞ is a monotone decreasing function on t. t Therefore, the result of the theorem holds. h For general k, we have
Ifk ðDÞ ¼ log
nt n þ 2 þ ðn 2Þðt þ 1Þk kðn 2Þðt þ 1Þk logðt þ 1Þ : t nt n þ 2 þ ðn 2Þðt þ 1Þk
The extremal values of I f k ðDÞ are much complicated, and we can not easily say exactly which graph attains the maximum value or the minimum value of I f k ðDÞ. As an example, in Fig. 3, we plot the values of I f r ðDÞ for n ¼ 104 ; k ¼ 10 and 1 6 t 6 n 2. 3.3. Extremal values for IV ðDÞ and IE ðDÞ The following entropy was introduced by Emmert-Streib and Dehmer in [43,44], which was called the vertex entropy. Let G be a generalized tree with hight h. Denote by jVj and jV i j the total number of vertices and the number of vertices on the ith level, respectively. A probability distribution of G is assigned as follows:
pVi ¼
jV i j : jVj 1
ð8Þ
Then the entropy of a generalized tree G is defined by
IV ðHÞ :¼
h X pVi logðpVi Þ: i¼1
Similarly, the edge entropy was also introduced in [43,44]. Denote by jEj and jEi j the total number of edges and the number of edges on the ith level, respectively. A probability distribution of G is assigned as follows:
pEi ¼
jEi j : jEj 1
Then the entropy of a generalized tree G is defined by
IE ðHÞ :¼
h X pEi logðpEi Þ: i¼1
Fig. 3. The values of entropies I f k ðDÞ for n ¼ 104 and k ¼ 10.
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Theorem 2. Let D be a dendrimer on n vertices. The star graph attains the minimal value of IV ðDÞ, and the dendrimer with parameter t ¼ t 0 attains the maximal value of IV ðDÞ, where t ¼ t 0 2 ð1; n 2Þ is the integer which is closest to the root of the equation
n nt n þ 2 tðt þ 1Þ 2t ln ln ¼ 0: n1 tþ1 n1 tþ1 Proof. From the definition of dendrimer, we have
pVi ¼
jV i j ðt þ 1Þt i1 ¼ : n1 n1
ð9Þ
Therefore,
r X ðt þ 1Þt i1 ðt þ 1Þti1 tþ1 ¼ þ ði 1Þ log t log n1 n1 n1 n1 i¼1 i¼1 ! r r X tþ1 t þ 1 X i1 log t þ log t ði 1Þt i1 ¼ n1 n 1 i¼1 i¼1 r1 tþ1 t þ 1 tr 1 t þ 1 tðt 1Þ log log t ðr 1Þt r : ¼ n1 n1 t1 n1 t1
IV ðGÞ ¼
r X ðt þ 1Þti1
log
ð10Þ
By substituting Eqs. (4) and (6) into (10), we have
tþ1 t þ 1 ðn 1Þðt 1Þ log n1 n1 t1 0 0 1 1 ntnþ2 log tþ1 tþ1 t nt t þ 2 nt n þ 2 A 1 @ log t @ 1A n1 t 1 tðt þ 1Þ tþ1 log t tþ1 nt2 nt n þ 2 nt n þ 2 ¼ ðt þ 1Þ log : log log t þ n1 n1 n1 tðt þ 1Þ
IV ðGÞ ¼
ð11Þ
By some elementary calculations and simplifications, we have
@ IV ðGÞ @t
1 ln t n nt n þ 2 tþ1 2t ln ln þ ln 2 n 1 n 1 tðt þ 1Þ n1 tþ1 1 n nt n þ 2 tðt þ 1Þ 2t ¼ ln ; ln ln 2 n 1 tþ1 n1 tþ1
¼
and
@ 2 ðIV ðGÞÞ ð3n2 2nÞt3 þ ð2n2 þ 4n 4Þt 2 ð7n2 16n þ 8Þt þ 2ðn 1Þðn 2Þ ¼ : 2 @t ðn 1Þðt þ 1Þ2 ðnt n þ 2Þt ln 2 Observe that
@ 2 ðIV ðGÞÞ @t 2
< 0 for all n and t. Then IV ðGÞ is a concave function on t. From Eq. (10), we know that IV ðGÞ ¼ 0
when t ¼ n 2, i.e., IV ðSn Þ ¼ 0. So, the star graph attains the minimal value of IV ðDÞ. On the other hand, the maximal value of IV ðDÞ will be attained when
n nt n þ 2 tðt þ 1Þ 2t ln ln ¼ 0: n1 tþ1 n1 tþ1
@ðIV ðGÞÞ @t
¼ 0, i.e.,
Therefore, the dendrimer with t ¼ t0 attains the maximal value of IV ðDÞ, where t ¼ t0 2 ð1; n 2Þ is the integer which is closest to the root of the equation
n nt n þ 2 tðt þ 1Þ 2t ln ln ¼ 0: n1 tþ1 n1 tþ1
Fig. 4 shows the relation between the values of n and t 0 . From Fig. 4, we can see that the exact value of t 0 is around 0:37n. Observe that for trees, the values of the vertex entropy and the edge entropy are equal. Then we have the following theorem for the edge entropy. Theorem 3. Let D be a dendrimer on n vertices. The star graph attains the minimal value of IE ðDÞ, and the dendrimer with t ¼ t 0 attains the maximal value of IE ðDÞ, where t ¼ t 0 2 ð1; n 2Þ is the integer which is closest to the root of the equation
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n nt n þ 2 tðt þ 1Þ 2t ln ln ¼ 0: n1 tþ1 n1 tþ1 3.4. Extremal values for the generalized graph entropies In [45], Dehmer and Mowshowitz introduced a new class of measures (called here generalized measures) that derived from functions such as those defined by Rényi’s entropy [46] and Daròczy’s entropy [47], which is now called the generalized graph entropies. We state the definition as follows. Let G be a graph of order n and X be a set of elements of a graph. Let s be an equivalence relation defined on X. Denote by X i and k the equivalence class and the number of equivalence classes of the relation s. Then,
jX i j ; jXj jXj i¼1 a ! n X 1 jX i j 2 ðiiÞ Ia ðGÞ :¼ log ; a – 1; 1a jXj i¼1 Pn jXi j a 1 i¼1 jXj 3 ; a – 1: ðiiiÞ Ia ðGÞ :¼ 1a 2 1
ðiÞ I1 ðGÞ :¼
n X jX i j
1
According to Rashevsky [18], jX i j denotes the number of topologically equivalent vertices in the ith vertex orbit of G, where k is the number of different orbits. Vertices are considered as topologically equivalent if they belong to the same orbit of a jV i j graph G. Now we suppose X ¼ jVj 1, and then the probability of X i can be expressed as pVi ¼ jVj1 , which is the same as
Eq. (8). Therefore, the entropies can be expressed as follows:
ðiÞ I1 ðGÞ :¼
n X pVi 1 pVi ; i¼1
! n X V a 1 ; a – 1; log pi ðiiÞ Ia ðGÞ :¼ 1a i¼1 Pn V a p 1 ; a – 1: ðiiiÞ I3a ðGÞ :¼ i¼11ai 2 1 2
Theorem 4. Let D be a dendrimer on n vertices. (i) The star graph and path graph attain the minimal and maximal value of I1 ðDÞ, respectively. (ii) For a – 1, the star graph and path graph attain the minimal and maximal value of I2a ðDÞ, respectively. (iii) For a – 1, the star graph and path graph attain the minimal and maximal value of I3a ðDÞ, respectively.
Fig. 4. The values n and t0 in Theorem 2.
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Proof. Pn
(i) Since I1 ðGÞ :¼
I1 ðGÞ ¼ ¼
V i¼1 pi
1 pVi , from Eq. (6), we infer
X 2 n n n X ðt þ 1Þt i1 ðt þ 1Þti1 ðt þ 1Þt i1 X ðt þ 1Þt i1 ðt þ 1Þðtr 1Þðn tr 2Þ ¼ ¼ 1 n 1 n 1 n 1 n 1 ðn 1Þ2 ðt 1Þ i¼1 i¼1 i¼1 2ðn t 2Þ : ðn 1Þðt þ 1Þ
ð12Þ
Thus, I1 ðDÞ is monotone decreasing on t, which implies that the star graph and path graph attain the minimal and maximal value of I1 ðDÞ, respectively.Now we are ready to show (ii) and (iii). From Eq. (9), we can obtain that
pVi ¼
ðt þ 1Þti1 : n1
Then from Eq. (6), we have r r X ðt þ 1Þa X ðt þ 1Þa ðt ar1 Þ ðnt n þ 2Þa ðt þ 1Þa a ¼ ðpVi Þ ¼ tði1Þa ¼ : a a a ðn 1Þ i¼1 ðn 1Þa ðta 1Þ ðn 1Þ ðt 1Þ i¼1
ð13Þ
By some elementary calculations and simplifications, we get
@
P r
V a i¼1 ðpi Þ
¼
@t
aðnt n þ 2Þa1 ðn 2Þta1 n þ aðt þ 1Þa1 ðta1 þ 1Þ ðn 1Þa ðt a 1Þ
2
:
Observe that nt n þ 2 P t þ 1, then ðnt n þ 2Þa1 P ðt þ 1Þa1 for a > 1 and ðnt n þ 2Þa1 6 ðt þ 1Þa1 for 0 < a < 1. Therefore, we have
@
P r
V a i¼1 ðpi Þ
P
@t
aðt þ 1Þa1 ðn 2Þta1 n þ aðt þ 1Þa1 ðta1 þ 1Þ a
a
ðn 1Þ ðt 1Þ
2
¼
aðn 1Þðt þ 1Þa1 ðta1 1Þ ðn 1Þa ðta 1Þ
2
>0
for a > 1, and
@
P r
V a i¼1 ðpi Þ
@t for 0 < a < 1. Then
6 Pr
ðnt n þ 2Þa1 ðn 2Þt a1 n þ aðnt n þ 2Þa1 ðt a1 þ 1Þ
V a i¼1 ðpi Þ
ðn 1Þa ðt a 1Þ
2
¼
aðn 1Þðnt n þ 2Þa1 ðta1 1Þ ðn 1Þa ðt a 1Þ
2
<0
is monotone increasing on t when a > 1 and monotone decreasing on t when 0 < a < 1.
(i) By the definition of I2a ðDÞ, we get that I2a ðDÞ is monotone decreasing on t for 0 < a < 1 and a > 1. Therefore, the star graph and path graph attain the minimal and maximal value of I2a ðDÞ, respectively.
Fig. 5. The values of entropies I f r ðDÞ (red), I f 1 ðDÞ (blue), IV ðDÞ (cyan), I1 ðDÞ (green), I2a ðDÞ (black) and I3a ðDÞ (grey), for a ¼ 0:5; t ¼ 2 (left) and t ¼ 3 (right). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 6. The values of entropies I f r ðDÞ (red), I f 1 ðDÞ (blue), IV ðDÞ (cyan), I1 ðDÞ (green), I2a ðDÞ (black) and I3a ðDÞ (grey), for a ¼ 2; t ¼ 2 (left) and t ¼ 3 (right). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
(ii) By the definition of I3a ðDÞ, we get that I3a ðDÞ is monotone decreasing on t for 0 < a < 1 and a > 1. Therefore, the star graph and path graph attain the minimal and maximal value of I3a ðDÞ, respectively.The proof is thus complete. h 4. Numerical Results In Fig. 5 and 6, we calculate the values of entropies I f r ðDÞ (red), I f 1 ðDÞ (blue), IV ðDÞ (cyan), I1 ðDÞ (green), I2a ðDÞ (black) and I3a ðDÞ (grey), considered in Section 3 for dendrimers with t ¼ 2 and t ¼ 3. From Fig. 5 and 6, we can see the change tendency of the value of each entropy according to the value of n. As we have seen from Fig. 5 and 6, for a ¼ 0:5 or a ¼ 2; t ¼ 2 or t ¼ 3, the following inequalities hold
IV ðDÞ > Ifr ðDÞ > If1 ðDÞ > I3a ðDÞ > I2a ðDÞ > I1 ðDÞ: On the other hand, the value of I f r ðDÞ is much larger than the other entropies. The difference between I f 1 ðDÞ and IV ðDÞ is not quite large. In general, the values of I1 ðDÞ; I2a ðDÞ and I3a ðDÞ are similar. In Fig. 2 and 3, we plot the values of I f r ðDÞ and I f k ðDÞ for n ¼ 104 and 1 6 t 6 n 2, respectively. As we have seen, the situation to determine the extremal values of I f r ðDÞ and I f k ðDÞ is complicated, and we can not easily say exactly which graph attains the maximum value or the minimum value of I f r ðDÞ and I f k ðDÞ, respectively. Simultaneously, according to the values
Fig. 7. The values of entropies I f 1 ðDÞ (left), IV ðDÞ (right) for n ¼ 104 .
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471
Fig. 8. The values of entropies I1 ðDÞ (left), I2a ðDÞ (middle), I3a ðDÞ (right) for a ¼ 0:5 and n ¼ 104 .
Fig. 9. The values of entropies I1 ðDÞ (left), I2a ðDÞ (middle), I3a ðDÞ (right) for a ¼ 2 and n ¼ 104 .
of t, we plot the values of I f 1 ðDÞ and IV ðDÞ for n ¼ 104 , respectively, as shown in Fig. 7. We also plot the values of I1 ðDÞ; I2a ðDÞ and I3a ðDÞ for n ¼ 104 and a ¼ 0:5 or a ¼ 2, respectively, as shown in Fig. 8 and 9. From Fig. 7–9, we can see that except IV ðDÞ, the star graph (t ¼ n 2) and path graph (t ¼ 1) attain the minimal and maximal value of entropies, respectively, which confirms the results of Theorems 1–4. 5. Conclusion In this paper, we tackled the problem of finding extremal values of graph entropies for dendrimers by using information functionals. This problem is highly relevant for designing new applications related to molecular structure networks, and also in the pharmaceutical and biomedical area. Among all dendrimers with n vertices, we obtain the extremal values of graph entropy based on several theorems (see previous section). From Theorems 1–4, we obtain that among all dendrimers with n vertices, the star graph (t ¼ n 2) and path graph (t ¼ 1) attain the minimal and maximal value of entropies, respectively. An exception is the entropy I f r ðDÞ and IV ðDÞ. The problem of determining the extremal values of I f r ðDÞ is intricate. It is interesting that, among all dendrimers with n vertices, the star graph attains the minimal value of IV ðDÞ, and the dendrimer with t ¼ t 0 attains the maximal value of IE ðDÞ, where t ¼ t 0 2 ð1; n 2Þ is the integer which is closest to 0:37n. Numerical experiments verified our mathematical findings. In the future, we would like to continue proving extremal results by employing graph entropies. In particular, it is interesting to investigate the extremal values of entropies for some other well-known network classes. A breakthrough would be to find analytical techniques to determine minimal graph entropies analytically for general graphs. The reason why this problem is still challenging relates to the fact that known results from information theory and majorization seem to be not applicable. Acknowledgments Zengqiang Chen was supported by the National Science Foundation of China (No. 61174094) and the Natural Science Foundation of Tianjin (No. 14JCYBJC18700). Matthias Dehmer and Yongtang Shi were supported by the Austrian Science Funds for supporting this work (Project P26142). Yongtang Shi was also are supported by NSFC, PCSIRT, China Postdoctoral Science Foundation (2014M551015) and China Scholarship Council.
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