# The Hosoya Entropy of a Graph

^{1}

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^{*}

## Abstract

**:**

## 1. Introduction

_{0}to B

_{10}range from 1 to 115, 975. Compared to the number of possibilities, it is clear that negligibly few entropy-based measures have been defined and studied so far. Thus, we contend that Hosoya entropy is a welcome addition to the small list inasmuch as it captures important structural features of a graph that have yet to be investigated from the perspective of complexity.

_{1}, …, v

_{n}}. Suppose further that d = d(G) is the diameter of G (i.e., the maximum distance between any pair of vertices in G), and that d(G, k) is the number of pairs of vertices in G at distance k from each other. Then $H(G,x):={\sum}_{k=1}^{d(G)}d(G,k){x}^{k}$.

_{i}is the cardinality of the i-th set of H-equivalent vertices for 1 ≤ i ≤ h, the Hosoya entropy (or H-entropy) of G (introduced in [19]) is given by

_{ij}), 1 ≤ i, j ≤ n of a connected graph G with n vertices, where d

_{ij}is the distance (or length of a shortest path) between vertices v

_{i}and v

_{j}. Thus, the respective numbers of elements in the i-th row equal to r for 1 ≤ r ≤ d are the coefficients of the partial Hosoya polynomial of v

_{i}. We will refer to these numbers, listed by distance, as the Hosoya profile or H-profile of v

_{i}. So, two vertices with the same H-profile, as found from the distance matrix, are H-equivalent.

_{k}be the number of entries in D(G) equal to k (0 ≤ k ≤ d). The measure in question is then the entropy of the finite probability scheme given by the d + 1 probabilities $\frac{{n}_{k}}{{n}^{2}}$. This measure is quite different from Hosoya entropy. Yet another related measure defined by Bonchev uses what he calls the Hosoya decomposition of a graph [3]. This decomposition is given by the family of matchings of orders zero through the maximum order max. Once again a finite probability scheme is constructed with elements n

_{k}/N where n

_{k}is the number of matchings of order k (0 ≤ k ≤ max) and N is the total number of matchings. The entropy of this scheme is called the information index on the Hosoya graph decomposition. This measure too is quite different from what we call Hosoya entropy.

_{4}(a path with 3 edges), illustrates the differences between these measures. The finite probability scheme for Bonchev’s measure based on the distance matrix is $({\scriptscriptstyle \frac{1}{4}},{\scriptscriptstyle \frac{3}{8}},{\scriptscriptstyle \frac{1}{4}},{\scriptscriptstyle \frac{1}{8}})$. The scheme for the measure based on matchings is $({\scriptscriptstyle \frac{1}{5}},{\scriptscriptstyle \frac{3}{5}},{\scriptscriptstyle \frac{1}{5}})$. Finally, the scheme for Hosoya entropy is $({\scriptscriptstyle \frac{1}{2}},{\scriptscriptstyle \frac{1}{2}})$.

## 2. Elementary Properties

**Theorem 1.**If G is a connected, r-regular graph with h ≤ 2, then H(G) = 0.

**Proof.**If its diameter is 1, G must be complete and thus has H-entropy 0. Suppose the diameter of G is 2. Since G is r-regular, the i − th row of its distance matrix consists of r 1′s and n − 1 − r 2′s, with a 0 in the i − th column, for 1 ≤ i ≤ n. So, all the rows of the distance matrix are permutations of each other, and hence the vertices are all H-equivalent. □

**Corollary 1.**If G is a regular, complete bigraph, H(G) = 0.

**Proof.**Since G is complete bipartite, its diameter is at most 2, and given that it is regular, G satisfies the conditions of the theorem. □

**Theorem 2.**If G is a connected, regular graph of degree > n/2, H(G) = 0.

**Proof.**Consider the product of any row of the distance matrix D(G) with any of its columns. Since G is regular of degree > n/2 each row (and column) of D(G) consists of at least n/2 1′s, and by the pigeon hole principle there must me at least one pair of 1′s in a common position which means the product must be positive. Thus, D

^{2}> 0 which means the diameter of G is at most 2, and by the previous theorem, H(G) = 0. □

^{n})

^{−1}for all n rows to be permutations of each other. Admittedly this value is far smaller than the actual probability since not all the permutations are consistent with the structural constraints of the graph. However, this approach provides a useful framework for investigating the problem.

## 3. Hosoya Entropy and Information Content

**Theorem 3.**Let G be a connected graph with automorphism group Aut(T). If vertices u and v of G are similar (i.e., belong to the same orbit of Aut(T)), they are H-equivalent.

**Proof.**Suppose u and v are similar, and let S(x) denote the Hosoya profile of vertex x. Let k

_{i}be an element of S(u), i.e, k

_{i}denotes the number of vertices at distance i from u, where 1 ≤ i ≤ h and h is the diameter of G. If S(u) ≠ S(v), then there is a k

_{i}∈ S(u) such that k

_{i}∉ S(v). Let ${w}_{1},\cdots ,{w}_{{k}_{i}}$ be the vertices whose distance from u is i. Then there is a path of length i from u to one of the w

_{j}that does not correspond to any path of length i from v, contradicting the similarity of u and v. □

**Corollary 2.**(1) If Aut(G) is transitive, then H(G) = 0.

**Proof.**(1) If Aut(G) is transitive, all the vertices of G are H-equivalent.

**Theorem 4.**There exist connected graphs with zero H-entropy whose automorphism groups are not transitive.

**Proof.**Let G be a regular n-vertex graph of degree 3 for which Aut(G) is not transitive. Such exists by a well known theorem of Frucht [20]. If n ≥ 7, the complement of G (which has the same automorphism group as G) is regular of degree > ${\scriptscriptstyle \frac{n}{2}}$, and by Theorem 2.2, $H(\overline{G})=0$. □

## 4. Hosoya Entropy of Trees

**Theorem 5.**(1) Let P

_{n}denote the path on n vertices, and let k ≥ 1. (a) H(P

_{n}) = log(k), if n = 2k, (b)$H({P}_{n})=\mathrm{log}(n)-{\scriptscriptstyle \frac{n-1}{n}}$, if n = 2k + 1.

_{n}denote the star for n ≥ 3. Then$H({S}_{n})=\mathrm{log}(n)-{\scriptscriptstyle \frac{n-1}{n}}\mathrm{log}(n-1)$.

**Proof.**(1) The H-equivalence partition of the vertices P

_{n}is the same as the orbit structure of its automorphism group. For n = 2k there are k equivalence classes of size 2; for n = 2k + 1 there are k equivalence classes of size 2 and one of size 1.

_{n}also enjoys the same H-equivalence partition as its orbit structure, namely one class of size n − 1 and one of size 1. □

_{2}. For level two, there are two subcases: both vertices are of degree one or two. Level three subdivides into three subcases: both vertices being of degree one, two, or three. For level greater than three, it is immediately evident that T must have at least ten vertices. In all the cases, it is obvious that either the two vertices cannot be H-equivalent or T has at least ten vertices.

**Theorem 6.**Vertices u and v in a tree T are similar if and only if the trees T

_{u}and T

_{v}rooted at u and v respectively are isomorphic.

**Proof.**Suppose the rootings of T, T

_{u}and T

_{v}rooted at u and v, respectively, are isomorphic, and that ϕ is the isomorphism mapping T

_{u}onto T

_{v}. Clearly, u and v are similar since ϕ can be seen as automorphism of T taking u to v. Conversely, let u and v be similar vertices of T, and suppose σ ∈ Aut(T) such that σ(u) = v. Without loss of generality we can assume that σ(u) = v, since T is a tree [21]. For any vertices x, y in T let [x, y] denote the unique path in T between x and y. Now, σ[u, x] = [σ(u), σ(x)] = [v, y] where σ(x) = y. Let x ≠ u be a vertex of T

_{u}. Consider the path [u, x] in T

_{u}. We have σ[u, x] = [σ(u), σ(x)] = [v, y]. Note that [v, y] is a path in T

_{v}. Since every edge in T

_{u}is on a path from u, and every edge in T

_{v}is on a path from v, then every edge ab in T

_{u}corresponds to the edge σ(a)σ(b) in T

_{v}, showing the two rootings to be isomorphic, which concludes the proof. □

## 5. Conclusion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Mowshowitz, A.; Dehmer, M. The Hosoya Entropy of a Graph. *Entropy* **2015**, *17*, 1054-1062.
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Mowshowitz, Abbe, and Matthias Dehmer. 2015. "The Hosoya Entropy of a Graph" *Entropy* 17, no. 3: 1054-1062.
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