Interrelations of graph distance measures based on topological indices.

In this paper, we derive interrelations of graph distance measures by means of inequalities. For this investigation we are using graph distance measures based on topological indices that have not been studied in this context. Specifically, we are using the well-known Wiener index, Randić index, eigenvalue-based quantities and graph entropies. In addition to this analysis, we present results from numerical studies exploring various properties of the measures and aspects of their quality. Our results could find application in chemoinformatics and computational biology where the structural investigation of chemical components and gene networks is currently of great interest.

Introduction

Methods to determine the structural similarity or distance between graphs have been applied in many areas of sciences. For example, in mathematics [1], [2], [3], in biology [4], [5], [6], in chemistry [7], [8] and in chemoinformatics [9]. Other application-oriented areas where graph comparison techniques have been employed can be found in [10], [11], [12]. Note that the terms ‘graph similarity’ or ‘graph distance’ are not unique and strongly depend on the underlying concept. The two main concepts which have been explored extensively are exact and inexact graph matching, see [13], [3]. Exact graph matching [2], [3] relates to match graphs based on isomorphic relations. An important example is the so-called Zelinka distance [3] which requires computing the maximum common subgraphs of two graphs with the same number of vertices. However, it is evident that this technique is computationally demanding as the subgraph graph isomorphism problem is NP-complete [14]. In contrast to this, inexact or approximative techniques for comparing graphs match graphs in an error-tolerant way, see [13]. A highlight of this development has been the well-known graph edit distance (GED) due to Bunke [15]. String-based techniques also fit into the scheme of approximative graph comparison techniques [1], [16]. This approach aims to derive string representations which capture structural information of the underlying networks. By using string alignment techniques, one is able to compute similarity scores of the derived strings instead of matching the graphs by using classical techniques. Concrete examples thereof can be found in [1], [16].

As mentioned, numerous graph similarity and distance measures have been explored. But in fact, there is still a lack of a mathematical framework to explore interrelations of these measures. Suppose let and be two comparative graph measures (i.e., graph similarity or distance measures) which are defined on the graph class . Typical questions in this idea group would be to prove interrelations of the measures by means of inequalities such as . For instance, inequalities involving graph complexity measures have been inferred by Dehmer et al. [17], [18].

The main contribution of this paper is to infer interrelations of graph distance measures. To the best of our knowledge, this problem has not been tackled so far when using graph distance measures. However, interrelations of topological indices interpreted as complexity measures have been studied, see [7], [19], [20], [17], [18]. For instance, Bonchev and his co-workers investigated interrelations of branching measures by means of inequalities [7], [19], [20]. Dehmer [17] examined relations between information-theoretic measures which are based on information functionals and between classical and parametric graph entropies [18]. We here put the emphasis on graph distance measures which are based on so-called topological indices. These measures themselves have not yet been studied. Note that we only consider distance measures (without loss of generality) as they can be easily transformed into graph similarity measures [21]. In order to define these measures concrete, we employ an existing distance measure (see Eq. (6)) and the well-known Randić index [22], the Wiener index [23], eigenvalue-based measures [24], and graph entropies [17], [25]. Also, we discuss quality aspects of the measures and state conjectures evidenced by numerical results.

Methods and Results

Topological Indices and Preliminaries

In this section, we introduce the topological indices which are used in the paper. A topological index [23] is a graph invariant, defined by

Simple invariants are for instance the number of vertices, the number of edges, vertex degrees, degree sequences, the matching number, the chromatic number and so forth, see [26].

We emphasize that topological indices are graph invariants which characterize its topology. They have been used for examining quantitative structure-activity relationships (QSARs) extensively in which the biological activity or other properties of molecules are correlated with their chemical structures [27]. Topological graph measures have also been applied in ecology [28], biology [29] and in network physics [30], [31]. Note that various properties of topological graph measures such as their uniqueness and correlation ability have been examined too [32], [33].

Suppose is a connected graph. The distance between the vertices and of is denoted by . The Wiener index of is denoted by and defined by

The name Wiener index or Wiener number for the quantity defined is common in the chemical literature, since Wiener [34] in 1947 seems was the first who considered it. For more results on the Wiener index of trees, we refer to [35].

In 1975, Randić [36] proposed the topological index ( and ) by using the name branching index or connectivity index, suitable for measuring the extent of branching of the carbon-atom skeleton of saturated hydrocarbons. Nowadays this index is also called the Randić index. In 1998, Bollobás and Erdös [37] generalized this index by replacing by any real number , which is called the general Randić index. In fact, the Randić index and the general Randić index became the most popular and most frequently employed structure descriptors used in structural chemistry [38]. For a graph , the Randić index of has been defined as the sum of over all edges of , i.e.,

where is degree of a vertex of . The zeroth-order Randić index due to Kier and Hall [6] is

For more results on the Randić index and the zeroth-order Randić index, we refer to [39], [22], [38].

For a given graph with vertices, are the eigenvalues of . The energy of a graph , denoted by , has been defined by

due to Gutman in 1977 [40]. For more results on the graph energy, we refer to [41], [24], [42].

Novel Graph Distance Measures

Now we define the distance measure [21]

which is a mapping . Obviously it holds , , and . In order to translate this concept to graphs, we employ topological indices and obtain

Further we infer a relation between the maximum value of and the extremal values of .

Observation 1

Let be a class of graphs. Suppose , then are the two graphs attaining the maximum value of if and only if are the graphs attaining the maximum and minimum value of , respectively.

Proof. Let , then is a monotone increasing function on . Therefore, the maximum value of is attained if and only if the maximum value of is attained.

From Observation 1 and some existing extremal results of topological indices, we obtain some sharp upper bounds of for some classes of graphs. As an example, we list some of those results for trees.

Theorem 1

Let and be two trees with vertices. Denote by and the star graph and path graph with vertices, respectively.

. The maximum value of is attained when and are and , respectively.

. The maximum value of is attained when and are and , respectively.

. The maximum value of is attained when and are and , respectively.

. The maximum value of is attained when and are and , respectively.

Interrelations of Graph Distance Measures

Observe that , which implies that . Some trivial properties of are as follows. Let be a class of graphs and . We get

However, is not a metric graph distance measure, since the triangle inequality for , does not hold generally. Actually, we obtain a modified version of the triangle inequality.

Theorem 2

Let be a topological index. Let be a class of graphs and . If

then we have .

Proof. We now suppose , since the proof of the other case is similar.

From the inequality , we get

Since , together with Eq. (12), we have

Therefore, we have the following inequality,

i.e., .

We emphasize if the Inequalities 11 are satisfied, the modified triangle inequality holds. In practice, the triangle inequality may not be absolutely necessary (e.g., for clustering and classification problems) and is often required to prove properties of the measures.

Theorem 3

Let and be two topological indices. Let be a class of graphs and . If

then

where is a constant.

Proof. Since

we obtain

Thus

i.e.,

Thus,

The proof is complete.

Suppose is also a topological index. Then if

we derive similarly

where is a constant. Therefore, we obtain the following theorem.

Theorem 4

Let and be three topological indices. Let be a class of graphs and . If

then we infer

where are constants.

Theorem 5

Let and be two topological indices. Let be a class of graphs and . If

then we get

where is a constant.

Proof. Since

we infer

And therefore,

Hence,

From the definition of , i.e.,

we obtain that

Finally, by substituting (35) into (33), we get the desired result.

Suppose is also a topological index. Then if

we have

where is a constant. Therefore, we obtain the following theorem.

Theorem 6

Let and be three topological indices. Let be a class of graphs and . If

then we have

and

where are constants.

Theorem 7

Let and be three topological indices. Let be a class of graphs and . If

then we infer

Proof. Since

we derive

And therefore,

i.e., . Hence we obtain

which implies that

By substituting (35) into (47), we easily obtain the assertion of the theorem.

By performing a similar proof as in Theorem 7, we obtain a more general result.

Theorem 8

Let be topological indices. Let . If

we infer

Theorem 9

Let and be three topological indices. Let be a class of graphs and . If

where , then we get

Proof. Since

we derive

Therefore,

which implies

By applying the substitutions

and

into (56), we obtain the final result.

By performing a similar proof as in Theorem 9, we obtain a more general result again.

Theorem 10

Let be topological indices. Let be a class of graphs and . If

where for , then we infer

Graph Distance Measures Based on Randić Index

In this section, we consider the values of the graph distance measure based on the Randić index and other topological indices for some classes of graphs. Denote by and the Wiener index and Randić index, respectively.

Theorem 11

Let be a class of regular graphs with vertices and is an arbitrary topological index. For two graphs , we infer

Proof. Let and be two regular graphs of order . By the definition of the Randić index, we obtain that , which implies that . Therefore, we infer . Since for any topological index, then we obtain the desired inequality.

By using the definition of the zeroth-order Randić index for two graphs with the same degree sequences, we obtain that . Therefore, we get the following theorem.

Theorem 12

Let be a class of graphs with the same degree sequences and is an arbitrary topological index. Then for two graphs , we infer

For a given graph of order , we get (see [39]). Thus,

From (63), we infer an upper bound for .

Theorem 13

Let and be two connected graphs of order . Then we get

The equality holds if and only if and are and a regular graph, respectively.

A path is pendent if , and for all . Especially, a vertex is pendent if . Suppose and are two pendent vertices, and the unique neighbor of . We define an operation as follows: deleting the edge and adding the edge . We call this operation “transfer to ”.

Theorem 14

Let be a graph with vertices. Denote by and the two pendent paths attaching to the same vertex such that . Denote by the graph obtained by transferring the pendent vertex of to the pendent vertex of . Then we have

Proof. Let be a graph with vertices. Suppose and with . Since and are two pendent paths attaching to the same vertex, then we get

By using the definition of , we infer . By using the definition of , we only need to show

Observe that . We will discuss the difference of the distances between two vertices in and . Let and be two vertices of . If , then we have . Now we suppose . If , then

Observe that

Therefore, we have

i.e,

For , it is easy to verify . Therefore holds.

For , from (66), we have and . By performing some elementary calculations, we get

i.e.,

for and each value of . Therefore, from (63), we infer .

For , from (66), we have and . By performing some elementary calculations, we obtain

i.e.,

for and each value of . Therefore, from (63), we infer . The proof is complete.

This theorem can be used to compare the values of the distance measure by using trees. Let be the set of trees with vertices and

Observe that for every , there must be a tree such that can be obtained from by repeatedly transferring pendent vertices. Therefore, we obtain the following corollary.

Corollary 1

Let , there exists a tree such that .

Actually, numerical experiments show that for any two trees , the inequality holds. We state the result as a conjecture.

Conjecture 1

Let and be any two trees with vertices. Then

holds.

As an example, we consider (all) trees with 8 vertices and calculate all possible values of (blue) and (red) as shown in Figure 1. From Figure 1, we observe that holds for each pair of trees and .

Figure 1

All the values of (blue) and (red).

The Y-axis denotes the values of the distance measure and the X-axis denotes the graph pairs.

Graph Distance Measures Based on Graph Entropy

In this section, we consider graph distance measures which are based on graph entropy and other topological indices for some classes of graphs.

In order to start, we reproduce the definition of Shannon's entropy [43]. Let be a probability vector, namely, and . The Shannon's entropy of has been defined by

We denote by the graph distance measure based on .

In the following, we infer an upper bound for .

Theorem 15

Let and be two graphs with the same vertex set. Denote by and be the probability vectors of and , respectively. If for each , then we infer

where .

Proof. Since for each , then we obtain and . Then we have

Therefore, we get the inequality,

i.e., . Hence,

The desired inequality holds.

In [25], Dehmer and Mowshowitz generalized the definition of graph entropy by using information functionals. Let be a connected graph. For a vertex , we define

where represents an arbitrary information functional. By substituting to (78), we have

We denote by the graph distance measure based on .

Relations between and

Denote by the eigenvalues of a graph . By setting in (87), we obtain a new expression of the graph entropy namely

Recall that the energy of is defined as . Then we infer

From the definition of , it is interesting to investigate the relation between the graph distance measures and .

Theorem 16

Let and be two graphs of order with . Denote by and the eigenvalues of and , respectively. Let and . Then we get

where is a constant.

Proof. Let and be two graphs of order . Let and with . Then we get

where . Thus,

i.e.,

Taking logarithm for the two sides of the above inequality, we have

The required inequality holds.

Actually, numerical experiments show that for any two distinct trees , holds. See Figure 2 as an example, in which we consider (all) trees with 8 vertices and calculate all possible values of (red) and (blue). We state this observation as a conjecture.

Figure 2

Values of (red) and (blue).

The Y-axis denotes the values of the distance measure and the X-axis denotes the graph pairs.

Conjecture 2

Let and be any two distinct trees with vertices. Then

holds.

Using a similar proof method of Theorem 16, we can obtain a generalization for the distance measure based on (see Eq. (87)). Let be an arbitrary information functional and be a topological index.

Theorem 17

Let and be two graphs of order with . Let and . Then we have

where is a constant.

Dehmer and Mowshowitz [44] introduced a new class of measures (called here generalized measures) that derive from functions such as those defined by Rényi's entropy and Daròczy's entropy. Let be a graph of order . Then

If we let , then we can obtain the new generalized entropy based on eigenvalues. We denote the entropy by

For a given graph with vertices, denote by the eigenvalues of . By substituting into equality (104), we have

The last equality holds since . By the following theorem, we study the relation between and .

Theorem 18

Let be a class of graphs with vertices and edges. For two graphs , let and . Then we get

and

where is a constant.

Proof. Let and be two graphs with vertices and edges. Without loss of generality, we suppose .

To show the first inequality, it suffices to prove

Then from (107), we derive

If we want to prove

we only need to show

From a well-known bound of energy , we have and . Therefore, holds.

Now we show the second inequality. From (111), we have

Therefore, we have

From the definition of the distance measure, by some elementary calculations, we finally infer

where is a constant.

The proof is complete.

Let be a connected graph with vertices, edges and degree sequence , where for . By setting in (87), we can obtain the new entropy based on degree powers, denoted by

For , the expression is just the zeroth-order Randić index . Then by using Theorem 17, we obtain the following result.

Theorem 19

Let and be two graphs of order with . Let

Then we have

where is a constant.

For , we get

Furthermore, by the definition of , for two graphs with the same degree sequences, we obtain that . Therefore, we get the following result.

Theorem 20

Let be a class of graphs with the same degree sequences and is an arbitrary topological index. Then for two graphs , we infer

By using the similar proof method applied in Theorem 14, we obtain a weaker result.

Theorem 21

Let be a tree with vertices. Denote by and two pendent paths attaching to the same vertex such that . Denote by the tree obtained by transferring the pendent vertex of to the pendent vertex of . Then we have

Proof. Let be a tree with vertices. Suppose and with . Denote by the degree of , i.e., . Since and are two pendent paths attaching to the same vertex, then we have . By using the definition of , we have . By using the definition of , we only need to show

For a tree with vertices, we get . By performing elementary calculations, we get

Observe that . We first discuss the difference of the distances between two vertices in and . Let and be two vertices of . If , then we have . Now we suppose . If , then Observe that

Therefore, we get

For , it is easy to verify that , i.e., . Then,

In the following, we suppose .

We obtain and . By performing elementary calculations, we get

for and each value of . Therefore,

To prove the other inequality, we need more detailed discussion. By using the definition of graph entropy, we get

Let be the set of the neighbors of vertex , which does not contain and . Denote by the degree of a vertex in , where . If , then

By performing some calculations, we can show that for and ,

i.e., for . For smaller , we verify this inequality directly. If , then we have

We can show that for and ,

i.e., for . For smaller , we verify this inequality directly. Now suppose , then there is only one vertex in whose degree is at most . Therefore by using (133) and (136), we get

and

We can verify

for each , i.e., .

From Theorem 14 and 21, we obtain the following corollary.

Corollary 2

Let be a tree with vertices. Denote by and the two pendent paths attaching to the same vertex such that . Denote by the tree obtained by transferring the pendent vertex of to the pendent vertex of . Then we have

Therefore, we obtain a similar result to comparing the values of distance measures of trees.

Corollary 3

Let , there exists a tree such that .

Actually, our numerical results (see section ‘Numerical Results’) show that for any two trees , the following inequality may hold.

Conjecture 3

Let and be any two trees with vertices. Then

holds.

By way of example, we consider all trees of 8 vertices and calculate all possible values of (blue) and (red), respectively, as shown in Figure 3. From Figure 3, we observe that

Figure 3

Values of (blue) and (red).

The Y-axis denotes the values of the distance measure and the X-axis denotes the graph pairs.

holds for each pair of trees and .

Numerical Results

In this section, we interpret the numerical results. First, we consider all trees with vertices. The number of trees is and the number of pairs is (see [45]). From the curves shown by Figure 1, we see that both measures (blue) and (red) satisfy the inequality Eq. (77). From the curves shown by Figure 2, we observe that both measures (red) and (blue) satisfy the inequality Eq. (101). From the curves shown by Figure 3, we also learn that both measures (blue) and (red) fulfill the inequality Eq. (143). By using this method, several other inequalities could be generated and verified graphically.

Figures 4 and 5 show the numerical results by using the graph distance measures based on graph energy , the Wiener index and the Randić index , respectively. We consider all trees with vertices. The number of trees is and the number of pairs is (see [45]). By Figure 4, we depict the distributions of the ranked distance values, that is, (red), (blue), and (yellow). First and foremost, we see that the measured values of all three measures cover the entire interval . This indicates that the measures are generally useful as they are well defined. By considering , we observe that only a relatively little number of pairs have a measured value 0.8. But a large number of pairs possess distance values 0.8. When considering , the situation is reverse. The distance values of seem to slightly increase with some up- and downturns. However, Figure 4 does not comment on the ability of the graph distance measures to classify graphs efficiently. This needs to be examined in the future and would far beyond the scope of this paper.

Figure 4

Distributions of the ranked values of the distance measure (red), (blue), (yellow).

The X-axis denotes the values of the distance measure. The Y-axis denotes the number of graph pairs.

Figure 5

The X-axis denotes the values of the distance measures (red), (blue), (yellow).

The Y-axis represents the percentage rate of all graphs studied.

Furthermore, we have computed the cumulative distributions by using the measures (red), (blue), (yellow), respectively, as shown in Figure 5. In general, the computation of the cumulative distribution may serve as a preprocessing step when analyzing graphs structurally. In fact, we see how many percent of the 235 graphs have a distance value which is less or equal . Also, Figure 5 shows that the value distributions are quite different. From Figure 5, we see that the curve for strongly differs from and . When considering , we also observe that about 80% of the trees have a distance value approximately 0.5. That means most of the trees are quite dissimilar according to . For , the situation is absolutely reverse. Here 80% of the trees have a distance value approximately 0.98. Finally evaluating the graph distance measure on these trees reveals that about 80% of the trees possess a distance value approximately 0.85. In summary, we conclude from Figure 5 that all three measures capture the distance between the graphs quite differently. But nevertheless, this does not imply that the quality of one measure may be worse than another. Again, an important issue of quality is fulfilled as the measures turned out to be well defined, see Figure 4. Another crucial issue would be evaluating the classification ability which is future work.

Summary and Conclusion

In this paper, we have studied interrelations of graph distance measures which are based on distinct topological indices. In order to do so, we employed the Wiener index, the Randić index, the zeroth-order Randić index, the graph energy, and certain graph entropies [25]. In particular, we have obtained inequalities involving the novel graph distance measures. Evidenced by a numerical analysis we also found three conjectures dealing with relations between the distance measures on trees.

From Theorem 1, we see that the star graph and the path graph maximize among all trees with a given number of vertices, for any topological index we considered here. Actually, this also holds for some other topological indices, such as the Hosoya index [46], [47], the Merrifield-Simmons index [48], [49], [47], the Estrada index [50], [51], [52], and the Szeged index [53], [54]. All other theorems we have proved in this paper shed light on the problem of proving interrelations of the measures. We believe that such statements help to understand the measures more thoroughly and, finally, they are useful to establish new applications employing quantitative graph theory [55]. We emphasize that the star graph and the path graph are apparently the two most dissimilar trees among all trees. Similar observations can also be obtained for unicyclic graphs or bicyclic graphs. Therefore, in the future, we would like to explore which classes of graphs have this property, i.e., identifying graphs (such as the path graph and the star graph) which maximize or minimize .

Another direction for future work is to compare the values of where are general graphs. For example, we could assume that and are obtained by only one graph edit operation, i.e., GED() = 1, see [15]. Then, all the graph which fulfill this equation are (by definition) similar. This construction could help to study the sensitivity of the measures thoroughly. Note that similar properties of topological indices have already been investigated, see [56]. As a conclusive remark, we mention that dynamics models on spatial graphs have been studied by Perc and Wang and other researchers, see [57], [58]. It would be interesting to study the distance measures in this mathematical framework as well.

CSV file containing descriptor values of 235 trees by using the Randić index.

(CSV)

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CSV file containing descriptor values of 235 trees by using graph energy.

(CSV)

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CSV file containing descriptor values of 235 trees by using the Wiener index.

(CSV)

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