Applied Mathematics and Computation 250 (2015) 352–355

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

Discrimination power of graph measures based on complexzeros of the partial Hosoya polynomial

http://dx.doi.org/10.1016/j.amc.2014.10.0480096-3003/� 2014 Elsevier Inc. All rights reserved.

⇑ Corresponding author at: Department of Biomedical Computer Science & Mechatronics, UMIT, Eduard Wallnoefer Zentrum 1, A-6060 Hall in TyrolE-mail addresses: matthias.dehmer@umit.at (M. Dehmer), shi@nankai.edu.cn (Y. Shi), abbe@cs.ccny.cuny.edu (A. Mowshowitz).

Matthias Dehmer a,b,⇑, Yongtang Shi c, Abbe Mowshowitz d

a Department of Computer Science, Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, 85577 Neubiberg, Germanyb Department of Biomedical Computer Science & Mechatronics, UMIT, Eduard Wallnoefer Zentrum 1, A-6060 Hall in Tyrol, Austriac Center for Combinatorics and LPMC-TJKLC, Nankai University, Tianjin 300071, PR Chinad Department of Computer Science, The City College of New York, Convent Avenue at 138th Street, New York, NY 10031, USA

a r t i c l e i n f o a b s t r a c t

Keywords:Discrimination powerGraph measuresHosoya polynomial

In this paper we define novel graph measures based on the complex zeros of the partialHosoya polynomial. The kth coefficient of this polynomial, defined for an arbitrary vertexv of a graph, is the number of vertices at distance k from v. Based on the moduli of the com-plex zeros, we calculate novel graph descriptors on exhaustively generated graphs as wellas on trees. We then evaluate the uniqueness of these measures, i.e., their ability to distin-guish between non-isomorphic graphs. Detecting isomorphism for arbitrary graphsremains a challenging problem for which highly discriminating graph invariants are usefulheuristics.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

Various graph polynomials have proved useful in mathematical chemistry and related disciplines [1–4]. Counting poly-nomials are of particular importance, and the Hosoya polynomial [1] is a prime example. Other such polynomials countingmatchings, for example, are discussed in [4,2]. The seminal work of Hosoya on counting polynomials [1] triggered the devel-opment of novel topological indices (also called structural graph measures or just graph measures). Related work focuses onthe distribution of the zeros of graph polynomials. For example, the zeros of flow and chromatic polynomials have been stud-ied by Jackson [3] and Woodall [5]. They discuss regions or zero-free intervals of these polynomials. Results along these linescan in principle be used to characterize structural properties of graphs. Another prominent example is the characteristicpolynomial of the adjacency matrix of a graph [6]. The spectrum of a graph, made up of the roots of this polynomial, hasbeen investigated extensively in an effort to characterize graph structure [7,6]. Approximations of eigenvalues for differentclasses of graphs have been determined, and inequalities representing interrelations between eigenvalues and other graphinvariants have been established [7,6].

Dehmer and Ilic [8] investigated the zeros of Hosoya and distance polynomials (also called Wiener Polynomial) in the spe-cial cases of trees and cyclic graphs. They established various bounds on regions in the complex plane where all zeros ofthese polynomials lie. Also, they proved some of the bounds to be optimal for given classes of graphs. Further contributionsdealing with zeros of graph polynomials can be found in [9,8,4].

, Austria.

M. Dehmer et al. / Applied Mathematics and Computation 250 (2015) 352–355 353

This paper focuses on the interpretation of the zeros of graph polynomials as structural graph measures. Lovász and Pel-ikán [10] pioneered the work in this area by investigating the largest positive eigenvalue of a tree, which they showed to be ameasure of structural branching [11,12]. Randic [13] discussed various graph measures based on eigenvalues. For example,he surveyed contributions dealing with the largest and second largest eigenvalue of the adjacency matrix of a graph (see[14]). Also, spectral-based measures such as the sum of the positive eigenvalues, the sum of the moduli of the eigenvalues,the multiplicity of the zero eigenvalue have been investigated [13,15]. A more recent contribution towards this problem isdue to Dehmer et al. [16]. Defining structural graph descriptors based on the zeros of the so-called information polynomialdefined for given vertex probability distributions, they found these descriptors to have low degeneracy [16].

The objective of this paper is to explore an important property of the novel graph measures based on the zeros of the partialHosoya polynomial. This property relates to discrimination power (also called uniqueness) [17,18]. We evaluate the measureson sets of exhaustively generated graphs and find that they are good at discriminating between non-isomorphic trees. Also,the numerical results show the limitations of this approach. In related work, Dehmer et al. [18] investigated the discrimina-tion power of information-theoretic measures and compared the results with some of the well-known Molecular ID numbersdue to Randic [17,19]. Also, Dehmer et al. [20] discussed a computational procedure to find complete graph invariants which isbased on employing highly discriminating measures. Earlier contributions have been made by Konstantinova [21,22] Bonchevet al. [23,24] and Balaban [25].

2. Methods and results

2.1. Novel graph measures based on complex zeros

In this section, we define the novel graph descriptors based on the partial Hosoya polynomial. The partial Hosoya poly-nomial of a vertex v i in the graph G ¼ ðV ; EÞ is given by [26]

Hv i ðG; zÞ :¼Xv2Vv–vi

zdðv i ;vÞ; ð1Þ

where dðv i;vÞ is the distance (i.e., length of a shortest path) between the vertices v i and v. Solving the equation

Hv i ðG; zÞ ¼ 0; ð2Þ

yields the complex zeros zv i1 ; z

v i2 ; . . . ; zv i

kiwhich are not equal to zero. We infer ki < jV j by applying the well-known fundamen-

tal theorem of Algebra [27,28] stating that a complex polynomial

f ðzÞ :¼ akzk þ ak�1zk�1 þ a1zþ a0; ak – 0; ak 2 C; ð3Þ

with degree degðf Þ ¼ k has k complex zeros.The basic idea is to use the zeros of the partial Hosoya polynomial to characterize the structure of a graph. As noted above,

we aim to evaluate the discrimination power of these novel graph invariants.For the following, we define the novel graph measures by

M1ðGÞ :¼ jzv11 j þ jz

v12 j þ � � � þ jz

v1k1j

� �þ jzv2

1 j þ jzv22 j þ � � � þ jz

v2k2j

� �þ � � � þ jzv jV j

1 j þ jzv jV j2 j þ � � � þ jz

v jV jkjV jj

� �; ð4Þ

M2ðGÞ :¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijzv1

1 j þ jzv12 j þ � � � þ jz

v1k1j

qþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijzv2

1 j þ jzv22 j þ � � � þ jz

v2k2j

qþ � � � þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijzv jV j

1 j þ jzv jV j2 j þ � � � þ jz

v jV jkjV jj

q; ð5Þ

and

M3ðGÞ :¼ jzv11 j þ jz

v12 j þ � � � þ jz

v1k1j

� �log jzv1

1 j þ jzv12 j þ � � � þ jz

v1k1j

� �

þ jzv21 j þ jz

v22 j þ � � � þ jz

v2k2j

� �log jzv2

1 j þ jzv22 j þ � � � þ jz

v2k2j

� �

þ � � � þ jzv jV j1 j þ jz

v jV j2 j þ � � � þ jz

v jV jkVj

� �log jzv jV j

1 j þ jzv jV j2 j þ � � � þ jz

v jV jkjV jj

� �: ð6Þ

M1 is the sum of the sums of the moduli of all partial Hosoya polynomials Hv1 ;Hv2 ; . . . ;Hv jV j . M2 is the sum of the square rootsof the sums of the moduli of all partial Hosoya polynomials. M3 represents an entropy-like measure taking the sums of themoduli of all partial Hosoya polynomials into account. Strictly speaking, M3 is not an entropy of the form H :¼ �

Pixi logðxiÞ

asP

ixi ¼ 1 does not hold in general for Eq. 6. We note that general information measures have been discussed by Aczél andDaróczy [29].

2.2. Numerical results

This section presents the numerical results of applying the measures to certain classes of graphs. The data sets used are asfollows. Ti; 10 6 i 6 18 are sets of all non-isomorphic tress with i vertices. Similarly, Ni; i ¼ 9 is the set of all non-isomorphicgraphs with i vertices. We generated these graphs using Nauty [30] and implemented the graph measures Mi; 1 6 i 6 3 in R.

Table 1Exhaustively generated sets of non-isomorphic trees. jT10j ¼ 106; jT11j ¼ 235; jT12j ¼ 551; jT13 j ¼ 1301.

Measure T10 T11 T12 T13

ndv S ndv S ndv S ndv S

M1 0 1,000000 0 1,000000 0 1,000000 0 1,000000M2 0 1,000000 0 1,000000 0 1,000000 0 1,000000M3 0 1,000000 0 1,000000 0 1,000000 0 1,000000

Table 2Exhaustively generated sets of non-isomorphic trees. jT14j ¼ 3159; jT15j ¼ 7741; jT16 j ¼ 19320, jT17j ¼ 48629.

Measure T14 T15 T16 T17

ndv S ndv S ndv S ndv S

M1 0 1,000000 0 1,000000 0 1,000000 0 1,000000M2 0 1,000000 0 1,000000 0 1,000000 0 1,000000M3 0 1,000000 0 1,000000 0 1,000000 0 1,000000

Table 3Exhaustively generated sets of non-isomorphic trees and graphs. jT18j ¼ 123757; jN9 j ¼ 261080.

Measure T18 N9

ndv S ndv S

M1 2 0,999983 245117 0,061142M2 2 0,999983 226955 0,130707M3 2 0,999983 226854 0,131093

354 M. Dehmer et al. / Applied Mathematics and Computation 250 (2015) 352–355

Note that many other quantitative network measures have already been implemented with the aid of R; the correspondingR-package we have employed is called QuACN [31].

As in [18], we measure the degree of degeneracy by the quantity ndv. This is the number of non-distinguishable values,i.e., the number of graphs that cannot be distinguished structurally by their descriptor values [18,22]. Also, we compute the

quantity SGðMÞ :¼ jGj�ndvjGj developed by Konstantinova [22] to evaluate the degree of degeneracy of a measure M as well. Here,

M is a graph measure and G an arbitrary graph class.Consider Table 1. We see that the measures Mi; 1 6 i 6 3 do not produce any degeneracies when evaluating them on

T10; . . . ; T13. That means all measures discriminate the trees uniquely (ndv ¼ 0 and, hence, S ¼ 1). The same holds for theresults presented by Table 2. The first degeneracies occur when using T18 (see Table 3). We see that all the graph measuresproduce two degeneracies only, i.e., 2 graphs could not be discriminated when using Mi; 1 6 i 6 3.

By contrast, the measures cannot discriminate the graphs of N9 very well. Finally Table 3 shows that the discriminationpower of Mi is actually quite low. Considering the values obtained for S (Table 3), we see that the discrimination power of themeasures is less than 13%. The low discrimination power of the measures on N9 is due to the fact that graphs in general arestructurally more complex than trees. This result shows that the computation of all descriptors Mi result in quite similarnumerical values. A plausible explanation for this is that the resulting polynomials as well as the moduli of the zeros arequite similar.

3. Summary and conclusion

In this paper, we used the zeros of partial Hosoya polynomials to define graph invariants. The zeros of these polynomialsare often complex-valued, so we used their moduli to define the measures. This work is related to earlier findings of Balabanet al. [32] as they explored measures similar to M1 and M2; however, they used descriptors as variables. These measuresturned out to be useful discriminators. Our analysis revealed the limitations of the measures applied to arbitrary graphs.Although it is not possible to generalize about the discriminative power of the measures based on T9 alone, the results do sug-gest that the partial Hosoya polynomials and, in particular, the moduli of their zeros do not capture enough structural infor-mation to discriminate between graphs. Nevertheless, the measures do discriminate quite well on trees, and may be moreefficient to compute than standard isomorphism algorithms for trees. This is an open problem that we intend to examinein future work. Other structural measures such as special graph entropies are highly discriminative on exhaustively generatedgraphs (see [18]), so we also plan to examine ways of improving the discriminative power of the Hosoya based measures. Wehave restricted our analysis to rather small classes of graphs and trees because of combinatorial explosion. The number ofgraphs on n vertices increases dramatically with the value of n. For example, N9 contains 261080 non-isomorphic and

M. Dehmer et al. / Applied Mathematics and Computation 250 (2015) 352–355 355

connected graphs. Increasing the number of vertices by one to obtain N10 yields a set containing almost 12 million graphs,causing severe computational difficulties when calculating the measures and computing the non-distinguishable values. Itis simply not feasible to perform the calculations on sets of larger graphs.

Therefore we propose to improve the discriminative power of polynomial based measures in future work. For example,we might compare the results of the Hosoya based measure with other graph polynomials. Another interesting issue is toestablish extremal properties of the measures, i.e., determine which classes of graphs or trees produce minimal and maximalvalues. A more general open problem is to devise statistically valid methods that would allow for generalizing the discrim-inatory power of measures based on the results of computations on relatively small samples of graphs of varying size.

Acknowledgments

Matthias Dehmer thank the Austrian Science Funds for supporting this work (Project P26142). Yongtang Shi are sup-ported by NSFC, PCSIRT, China Postdoctoral Science Foundation (2014M551015) and China Scholarship Council. We thankLukas Huber for help regarding the data analysis and fruitful discussions.

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