Entropy measures of distance matrix

In this paper, two distance-based entropy measures are studied that were introduced by Bonchev and Trinajsti´c [ J. Chem. Phys. 38 (1977) 4517–4533] for interpreting the molecular branching of molecular graphs. One of these entropy measures is based on the distribution of distances in the distance matrix and the other one is based on the distribution of distances in the upper triangular submatrix. The mentioned measures are calculated for paths, stars, complete graphs, cycles, and complete bipartite graphs. The minimal trees with respect to the entropy measures under consideration with ﬁxed diameter are also determined.


Introduction
The entropy concept was introduced by Shannon in 1948 [17]. The Shannon entropy can be applied to different networks with the possibility of constructing a finite probability scheme for each network. The graph entropy concept was first defined by Rashevsky [15] in 1955. His entropy measure is based on the partition of the vertices with respect to equivalent classes of vertex degrees. Trucco [18] extended this definition by using automorphism groups of graphs. Mowshowitz [13] applied the information theory to different chemical structures and mathematical structures in 1968. Some properties of graph entropies were reported by Das and Shi [5]. Detail about the distance-based entropy measures can be found in [4,14]. The entropy measure based on dominating sets of graphs was introduced recently in [16]. More information about graph entropies can be found in the survey [7] and in the book [6].
Many molecular properties of materials are obtained by molecular topologies [11]. These measures are called topological indices or molecular descriptors in chemical graph theory. Many chemical, physical and biological properties of molecules have good correlations with several well-known topological indices. Therefore, many researchers from a wide range of sciences study on this topic. The first topological index was introduced by Wiener in 1947 [19]. The Wiener index is equal to one half of the sum of distances between every pair of vertices in a graph. The minimal trees of fixed diameter with respect to the Wiener index are characterized by Liu and Pan [12]. Detail about the Wiener index can be found in the excellent survey [8].
So many topological indices have been introduced in the last fifty years. It is understood that they have usually correlated more or less with the relative molecular properties of molecules but the same index can not has high discrimination ability for different molecules [11]. Bonchev and Trinajstić [1] introduced the entropy measures based on distances for interpreting the molecular branching of molecular graphs. Later, they applied the information theory in characterizing chemical structures [2,3]. These molecular descriptors were called information indices and it was shown that the information indices have greater discriminating power for molecules than the respective topological indices [9, 10].

Preliminaries
Let G be a simple graph with the vertex set V (G) and the edge set E(G). For a vertex u ∈ V (G), the notation N G (u) = {v | uv ∈ E(G)} denotes the vertices which are adjacent to u and N G [u] = {u} ∪ N G (u). The degree of a vertex u is the cardinality of N G (u) and it is denoted by deg G (u) or simply deg(u). A vertex which has degree one is called a leaf. Moreover, the distance between the vertices u and v is denoted by d(u, v). The maximum distance between any two vertices of a graph G is called diameter and is denoted by diam(G). The distance matrix D = [d ij ], (i, j = 1, 2, . . . , n) contains the distances d ij = d(i, j) between any two vertices of a connected graph.
The number of vertices of a graph G is called order and it is denoted by |V (G)| = n. The paths, cycles and stars of order n are denoted by P n , C n and S n , respectively. Moreover, complete graphs of order n are denoted by K n and complete bipartite graphs are denoted by K s,t with bipartite sets {u 1 , u 2 , . . . , u s } and {v 1 , v 2 , . . . , v t }. Definition 2.1 (see [17]). For a given probably vector p = {p 1 , p 2 , . . . , p n } such that 0 ≤ p i ≤ 1 and n i=1 p i = 1, the ShannonŠs entropy of p is presented by the following equation [19]). The Wiener index of a graph G is defined as

Definition 2.3 (see
We now can present the definitions of the information entropies introduced by Bonchev and Trinajstić [1]. In the distance matrix of a graph G, the distance i appears 2n i times, where 1 ≤ i ≤ diam(G). Thus, n 2 elements of the distance matrix of G are partitioned into diam(G) + 1 groups with n zeros which are diagonal elements of the matrix. Therefore, the probability distribution of the diam(G) + 1 groups for each i-th group is presented in Table 1 (see [11]), where p i = 2n i /n 2 for 1 ≤ i ≤ diam(G) and p 0 = n/n 2 = 1/n. Distance 0 1 2 · · · diam(G) Frequency n 2n 1 2n 2 · · · 2n diam(G) Probability 1/n p 1 p 2 · · · p diam(G) Since the distance matrix is symmetric, in order to simplify the calculations, only the upper triangular submatrix are be used. Thus, n(n − 1)/2 upper off-diagonal elements are used for the calculations. Definition 2.4 (see [1,11]). For a given distance i in a graph G such that 1 ≤ i ≤ diam(G), the information entropies I(G) and I * (G) are defined as follows .
In order to make some calculations, we order the diagonals of a square matrix We order the diagonals of A as follows: It is noted that if A is a symmetric matrix, then the following relations are hold: We note that diag(n) is the main diagonal of the matrix A and |diag(n)| = n. Moreover, the number of the elements which

Information entropies of some graphs
Theorem 3.1. The information entropies I and I * of the path graph P n of order n are given by the following formulas: Proof. The distance matrix of P n is presented as follows: It can be seen that the distance n − 1 appears in diag(1) and diag(2n − 1). The distance n − 2 appears in diag (2) and diag(2n − 2), and generally the distance n − i appears in diag(i) and diag(2n − i) with the frequency The diag(n) contains n zeros. From the matrix D(P n ), we obtain the frequency of distances (F req.), probability distributions of distances in the distance matrix (p i ) and probability distributions of distances in the upper triangular distance matrix (p * i ) as shown in Table 2. Table 2: Probability distributions for the path graph P n . i Now, by using the definitions of the information entropies I and I * , we obtain the required formulas.
Theorem 3.2. The information entropies I and I * of the star graph S n of order n are given by the following formulas, Proof. Let S n be a star of order n with the vertex set {v 1 , v 2 , . . . , v n } such that v n is the central vertex of S n . Then, the distance matrix of S n is presented as follows: Note that the distance 1 appears only in the n-th row and in the n-th column with 2n − 2 times. Therefore, the distance matrix D(S n ) consists of n zeros, 2n − 2 times 1, and n 2 − 3n + 2 times 2.
From the matrix D(S n ), we obtain the frequency of distances (F req.), probability distributions of distances in the distance matrix (p i ), and probability distributions of distances in the upper triangular submatrix (p * i ) as shown in Table  3. Table 3: Probability distributions for the star graph S n . i Now, by using the definitions of the information entropies I and I * , we obtain the I(S n ) and I * (S n ).
Theorem 3.3. The information entropies I and I * of the complete graph K n of order n are given by the following formulas: Proof. It is known that the distance matrix of the complete graph K n is consisted of n times 0 which are diagonal elements and n 2 − n times 1 which are the off-diagonal elements of the distance matrix D(K n ). Therefore, we obtain the probability distributions of K n as shown in Table 4. By using the definitions of the information entropies I and I * , we obtain the required formulas.
Generally, we obtain that 2n i = 2n for 1 ≤ i ≤ (n/2) − 1. We can investigate the frequency of the diam(C n ) = n/2. The distance n/2 appears in diag(n/2) in the upper triangular submatrix and in diag(3n/2) in the lower triangular submatrix. Therefore, we obtain From the matrix D(C n ), the probability distributions of distances are obtained as shown in Table 5. Table 5: Probability distributions for the cycle graph C n when n is even.
By using the definitions of the information entropies I and I * , we obtain I(C n ) and I * (C n ).
By the definitions of the information entropies I and I * , we obtain I(C n ) and I * (C n ).
Thus, from the D(K n 2 , n 2 ) matrix, the probability distributions of distances are obtained as shown in Table 7.

Some relations with respect to the information entropies I and I *
In order to make some comparisons, we use majorization method [5]. We consider non-increasing arrangement of each vector in R n . Consider a vector x = (x 1 , x 2 , . . . , x n ) ∈ R n such that x 1 ≥ x 2 ≥ · · · ≥ x n . Definition 4.1 (see [5]). For x, y ∈ R n , x ≺ y if When x ≺ y, x is said to be majorized by y (or y majorizes x).
Let p(G) = (p(v 1 ), p(v 2 ), . . . , p(v n )) be a probably vector of a graph G for the vertex set {v 1 , v 2 , . . . , v n } such that p(v 1 ) ≥ p(v 2 ) ≥ . . . ≥ p(v n ) and n i=1 p(v i ) = 1. Since the fuction h(x) = −x log x is a concave function for x > 0, we give an essential theorem as used in [5] for the information entropy If the diameter of two graphs are equal, then the distance matrices of these graphs are consisted of the same group distances but their frequencies can be different. It implies that the entropy measures I, I * can be compared by majorization method.
Lemma 4.1. Let G be a graph and u, v ∈ V (G). If G s,t is the graph obtained from G by attaching s, t leaves to the vertices v, u (respectively), the information entropies of the graphs G s,t , G s−i,t+i and G s+i,t−i (see Figure 1) are compared by the following inequalities with fixed diameter: Figure 1: The graph G s,t .
Proof. i) Assume that a leaf y is removed from v and it is attached to u. Therefore, the tree G s−1,t+1 is obtained. Let x = (. . . , p 2 (G s,t ), . . .) and x = (. . . , p 2 (G s−1,t+1 ), . . .) be nonincreasing probably vectors of G s,t and G s−1,t+1 . The total distance from y to other leaves which are incident to v is 2(s − 1). Moreover, the total distance from s − 1 leaves to y is also 2(s − 1). It means that if y is removed from v, the frequency of the distance 2 is decreased 2(s − 1) times in the distance matrix of G s,t . In the graph G s−1,t+1 , the frequency of the distance 2 is increased 2t times because of the leaf y is attached to u. Then, the difference of probabilities of the distance 2 in the distance matrices of G s−1,t+1 and G s,t is It implies that x ≺ x and I(G s−i,t+i ) ≤ I(G s,t ) or I(G s+i,t−i ) ≤ I(G s,t ).
ii) The same argument is used for the upper triangular submatrix and so the result is obtained.
Let P d+1 : v 0 v 1 . . . v d be a path of order d + 1. The graph which is obtained from P d+1 by attaching k = n − d − 1 leaves to i-th vertex of P d+1 is denoted by P d+1,i,k (see Figure 2). Figure 2: The graph P d+1,i,k .

Proof.
i) Assume that a leaf y which is attached to the i-th vertex of P d+1 is moved on the (i+1)-th vertex of P d+1 such that 1 < i+1 ≤ d 2 . Now, we order the distances from y to other vertices v 0 , v 1 , . . . , v d such that i+1, i, . . . , 1, 2, . . . , d−i+1 in P d+1,i,1 , respectively. Similarly, distances from y to other vertices v 0 , v 1 , . . . , v d are ordered i + 2, i + 1, . . . , 1, 2, . . . , d − i in P d+1,i+1,1 , respectively. It means that the frequencies of greater distances are decreased in the distance matrix and the frequencies of small distances are increased. This trend continues until the leaf y arrives to the d 2 -th vertex of the graph. Then, the probably vector of P d+1,i,1 is majorized by the probably vector of P d+1,i+1,1 and I(P d+1,i+1,1 ) ≤ I(P d+1,i,1 ). It can be generalized as I(P d+1,i,k ) ≤ I(P d+1,j,k ) for k leaves.
ii) It is obtained by the same way as used in the proof of item (i).

Conclusion
There are many open problems concerning the entropy measures considered in this paper. The extremal trees as well as the extremal unicyclic and bicyclic graphs can be obtained with respect to these entropy measures with different fixed parameters. It is observed that the distribution of vertex degrees of graphs is well studied but the distribution of distances is not studied well in the literature. The entropy measures considered in this paper are useful tools for the distribution of distances in a graph. Also, the average distance µ(G) of a graph G is defined (see [8]) as µ(G) = W (G) n(n − 1) .
The average distance µ related to the entropy measure I * . It seems to be an interesting problem to find relations between µ and I * .