PLoS ONEplosplosonePLoS ONE1932-6203Public Library of ScienceSan Francisco, USAPONE-D-13-3984010.1371/journal.pone.0083868Research ArticleComputer scienceAlgorithmsComputing methodsMathematicsApplied mathematicsDiscrete mathematicsMathematical computingSocial and behavioral sciencesInformation scienceInformation theoryThe Uniqueness of -Matrix Graph InvariantsThe Uniqueness of the $D_{ \textrm{MAX}}$-MatrixDehmerMatthias^{1}^{*}ShiYongtang^{2}Institute for Bioinformatics and Translational Research, UMIT, Eduard Wallnoefer Zentrum 1, A-6060, Hall in Tyrol, AustriaCenter for Combinatorics and LPMC-TJKLC, Nankai University, Tianjin, P.R. ChinaEmmert-StreibFrankEditorQueen's University Belfast, United Kingdom* E-mail: matthias.dehmer@umit.at

The authors have declared that no competing interests exist.

Wrote the paper: MD YS.

201421201491e83868249201391120132014Dehmer, ShiThis is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

In this paper, we examine the uniqueness (discrimination power) of a newly proposed graph invariant based on the matrix defined by Randić et al. In order to do so, we use exhaustively generated graphs instead of special graph classes such as trees only. Using these graph classes allow us to generalize the findings towards complex networks as they usually do not possess any structural constraints. We obtain that the uniqueness of this newly proposed graph invariant is approximately as low as the uniqueness of the Balaban index on exhaustively generated (general) graphs.

Matthias Dehmer thanks the Austrian Science Funds for supporting this work (project P22029-N13). Yongtang Shi has been supported by the National Science Foundation of China. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.Introduction

Matrix-based descriptors have been developed extensively [1]–[3]. As a result, the distance matrix, the adjacency matrix and other graph-theoretical matrices [4] have been used to define topological graph measures and to examine their properties [4], [5]. A property which has been of considerable interest when designing topological descriptors is referred to as uniqueness [6]–[8]. Generally, the uniqueness of a structural graph measure relates to the ability to distinguish the structure of non-isomorphic graphs uniquely. From a mathematical point of view, the low uniqueness or high degeneracy of a graph measure under consideration is an undesired aspect as non-isomorphic graphs should be mapped to non-equal values. Such a highly discriminating graph invariant could be then used to distinguish graph structures uniquely and, thus, to perform graph isomorphism testing [9], [10]. In the context of graph isomorphism testing, so-called complete graph invariants have been investigated [9], [11]. Such a graph invariant has the property that it discriminates all non-isomorphic graphs uniquely (i.e., without any degeneracy) and isomorphic graphs are mapped to equal values [9], [11]. For example, Liu and Klein [11] made an attempt to derive complete graph invariants by using eigenvalues. Dehmer et al. [8], [9] defined graph entropies which turned out to be the most discriminative measures so far when using exhaustively generated graphs. Clearly, such measures are suitable candidates to test graph isomorphism efficiently [9].

Recently, Randić et al. [2], [12] defined so-called matrices and also topological descriptors thereof. Let be a finite graph. Then these matrices have been defined by using the ordinary distance matrix of such that in each row and column the dominant (largest) distances are used where other elements are set to be zero, see [2], [12]. Moreover they defined a new topological index which has the same definition than the well-known Balaban index [13] but uses instead of only using . Then based on example claculations, Randić et al. [2] argued that may be a promising candidate for isomorphism testing, but they did not examine the problem in depth on wider classes of graphs.

In this paper, we explore the uniqueness of by employing on a large scale. For this, we use exhaustively generated graphs with 9 and 10 vertices each [8] and alkane trees where . Our findings (see section 'Methods and Results') reveal that the uniqueness of is always worse than the one of and, thus, the uniqueness of is insufficient for performing isomorphism testing.

Methods and ResultsThe Structural Descriptors <inline-formula><inline-graphic xlink:href="info:doi/10.1371/journal.pone.0083868.e021" xlink:type="simple"/></inline-formula> and <inline-formula><inline-graphic xlink:href="info:doi/10.1371/journal.pone.0083868.e022" xlink:type="simple"/></inline-formula>

Let be a finite graph. To define the Balaban index [13], [14] of , let be the distance matrix. is the topological distance between and . For each vertex , denotes the distance sum (row or column sum) by adding the entries in the corresponding row or column of . Let be the cyclomatic number [14]. Then has been defined by [13]

A critical analysis to examine the uniqueness of and other quantities has recently been carried out by Dehmer et al. [8] based on using exhaustively generated (general) graphs. In this sense Dehmer et al. examined the limitations of the Balaban index and found that this index is quite unstable [9]; here that means there is a strong dependency between the sample size of the graph set and the uniqueness [8]. To study the technical details and the precise definitions, we refer to [8], [9]. Moreover, the findings of Dehmer et al. [8] revealed that the uniqueness of by using exhaustively generated graphs is poor. For example by using the class (all non-isomorphic graphs with 10 vertices), , the Balaban index could only discriminate 20% of uniquely. Nevertheless, has high uniqueness for alkane trees and isomers [8], [9], [13].

To define , we require the definition of [12]:

Following Randić et al., the topological index is just the analog to Balaban's index, see [2]. Based on the fact that can discriminate the remaining isomers of -dodecane and has often a different structure compared to , Randić et al. concluded that and, hence, may be a promising tool for graph isomorphism testing, see [2]. In the next section, we see that this statement has been too premature when evaluating on general and exhaustively generated graphs. By evaluating characteristic properties (e.g., the uniqueness) of topological graph measures on such (general) graphs, one can conclude how would the index behave in the context of using complex networks.

Results

Before interpreting Table 1, we explain its notation. We here used the graph classes , [8] and , [8]. Again is the class of all exhaustively generated non-isomorphic and connected graphs with vertices [8]. is the class of exhaustively generated non-isomorphic and connected alkane trees [8]. ndv stands for the number of non-distinguishable values [8] and where is a class of graphs, see [7].

10.1371/journal.pone.0083868.t001The uniqueness of <inline-formula><inline-graphic xlink:href="info:doi/10.1371/journal.pone.0083868.e066" xlink:type="simple"/></inline-formula> and <inline-formula><inline-graphic xlink:href="info:doi/10.1371/journal.pone.0083868.e067" xlink:type="simple"/></inline-formula> measured by ndv and <inline-formula><inline-graphic xlink:href="info:doi/10.1371/journal.pone.0083868.e068" xlink:type="simple"/></inline-formula>.

Uniqueness of and

Graph class

cardinality

ndv

based on

261080

165109

0.36759

11716571

9476268

0.1912081

148284

144041

0.028614

366319

359177

0.019496

910726

898838

0.013053

2278658

2258804

0.00871302

261080

156674

0.399900

11716571

9307263

0.205633

148284

5967

0.95975

366319

44800

0.877702

910726

45703

0.949816

2278658

306911

0.865310

Table 1 shows numerical results when comparing and on the just explained graph classes. We observe that the uniqueness of is quite poor for all graph classes. In case of using , the uniqueness of is approximately as low as the uniqueness of . That means both topological indices can only discriminate about 39% out of 261080 graphs. By considering the results for , we see that possesses high uniqueness when using . Note that this has already been found by Balaban [13] and Dehmer et al. [8]. But it is surprising that the uniqueness of is, without exception, much worse than the one of . Table 2 shows that can discriminate the isomers of -dodecane for which the Balaban index is pairwisely degenerated.

10.1371/journal.pone.0083868.t002The values of <inline-formula><inline-graphic xlink:href="info:doi/10.1371/journal.pone.0083868.e101" xlink:type="simple"/></inline-formula> and <inline-formula><inline-graphic xlink:href="info:doi/10.1371/journal.pone.0083868.e102" xlink:type="simple"/></inline-formula> by using the isomers of n-dodecane.

Graph ID

1

4.252509

13.15875

2

4.252509

16.04085

3

3.752273

15.86058

4

3.752273

11.99851

5

4.135003

15.04953

6

4.135003

11.07033

7

3.575256

13.16886

8

3.575256

11.81169

9

3.773441

21.73837

10

3.773441

7.43385

11

3.954123

12.76649

12

3.954123

10.84469

A hypothesis is that the sparseness of leads to this effect described above. So this matrix can not capture the complexity of the used graphs meaningfully and, thus, is degenerated for most of the graphs. This result shows the complexity of the problem to construct highly unique graph measures on general and exhaustively generated graphs.

Summary and Conclusion

This paper investigated the uniqueness of the recently developed topological index introduced by Randić et al. [2]. has been defined quite similarly as it is based on the novel matrix instead of . Based on small tests and by only using example graphs, Randić et al. [2] hypothesized has higher uniqueness than and, the index which combined with index , may suffice to resolve the graph isomorphism issue for most cases of molecular graphs.

In this paper we have evaluated this hypothesis on a large scale by using general graphs. In fact, our study disproved this conjecture and demonstrated that the uniqueness of is quite poor by using general exhaustively generated graphs and alkane trees. As future work, we plan to determine so-called degeneracy classes analytically for performing a proper mathematical treatment of the problem. In any way, the search for highly discriminating graph invariants should be continued [8], [9], [15], [16]. Following Randić et al. [2], such measures could be used as a prescreening method and would eliminate need for detailed and elaborate tests on large number of cases. Also, this fact has already been raised by Dehmer et al. [9] where they developed information-theoretic network measures with very low degeneracy on exhaustively generated graphs for graph isomorphism testing.

We thank Shailesh Tripathi for help and fruitful discussions. We also thank the 'Zentraler Informatikdienst' of the Technical University of Vienna for providing computing resources to perform large scale computations.

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