From Sharma-Mittal to von-Neumann Entropy of a Graph

In this article, we introduce the Sharma-Mittal entropy of a graph, which is a generalization of the existing idea of the von-Neumann entropy. The well-known Rényi, Thallis, and von-Neumann entropies can be expressed as limiting cases of Sharma-Mittal entropy. We have explicitly calculated them for cycle, path, and complete graphs. Also, we have proposed a number of bounds for these entropies. In addition, we have also discussed the entropy of product graphs, such as Cartesian, Kronecker, Lexicographic, Strong, and Corona products. The change in entropy can also be utilized in the analysis of growing network models (Corona graphs), useful in generating complex networks.

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1 Introduction

“Graph entropy” was first introduced in [1, 2, 3] and applied for the problems in diverse fields to characterize the structure of graphs and to cater to the needs of an application. For instance, in mathematical chemistry, the graph entropy represents the structural information of graph-based systems [4], and the molecular structures [5]. This idea is utilized in finding the best possible encoding of messages, where the vertices of graphs are considered as symbols [6]. It is also a measure of the structural complexity in social networks [7, 8]. There are varieties of entropy functions on graphs defined in a different context [9, 10]. The relation between these entropy functions and the structural properties of graphs is a fundamental topic of research [11, 12]

. The eigenvalues of graphs provide an elegant tool in this context. In this article, we propose Sharma-Mittal entropy of graphs which is a generalization of a number of well-known eigenvalue based entropies.

This work is at the interface of quantum information theory, entropy, and graph theory. The graph Laplacian quantum states are represented by a Laplacian matrix of a graph. The von-Neumann entropy is the measure of quantum information in a quantum state, which has been considered as the entropy of graph. Later it was generalized as the Rényi and Tsallis entropy. The Sharma-Mittal entropy is generalization of all of them. It is observed that these generalized entropies are more efficient in quantum information theory problems than the von-Neumann entropy [13]. The Von Neumann entropy of graphs was introduced in[14] and then analyzed in a number of works [15, 16, 17, 18, 19, 20, 21, 22]. Three different Laplacian matrices of a graph are considered in the literature, which are the combinatorial Laplacian, signless Laplacian, and normalized Laplacian matrices. These articles primarily investigate the relation between the structure of the graph and the von-Neumann entropy. The relation between the quantum entanglement in graph Laplacian quantum states and the components of the graph is investigated in [23]. Random graphs are also recognized in the literature of the von-Neumann entropy of graphs [24]. The complexity of graphs is studied from the perspectives of thermodynamics in this direction [25].

To the best of our knowledge, this article is the first introduction of the Sharma-Mittal entropy based on graph Laplacian quantum states, in the literature. We extensively study the Sharma-Mittal entropy of the graphs and establish other entropies as its limiting cases. The Laplacian eigenvalues of cycle, path and complete graphs are known. We calculate the Sharma-Mittal entropy for them explicitly. Apart from calculating the entropies of the graphs, we attempt to provide a number of limiting values by calculating the upper and lower bounds for entropy functions. The product graphs, such as the corona product of graphs, are useful in modeling complex networks. In this work, we also calculate entropies for varieties of product graphs.

Our paper is organized as follows. The following section contains a very brief review of the graph Laplacian quantum states to define generalized entropy functions for graphs. Here we also calculate these entropies for a number of graphs with known spectra. We utilize combinatorial Laplacian and signless Laplacian for our investigation. In section 3, we construct a number of bounds on the entropies in terms of graph parameters. The section 4 is dedicated to the study of entropy in product graph and a network growing model. Then we conclude the article.

2 Graph Laplacian quantum states and generalized von-Neumann entropies

A graph consists of a set of vertices , and a set of unordered pair of vertices called edges. Throughout this article, and are the numbers of vertices, and edges of a graph , respectively. The adjacency matrix of a graph

 aij={1,if(vi,vj)∈E(G);0,if(vi,vj)∉E(G). (1)

The degree of a vertex is . The degree matrix of a graph denoted by . The combinatorial Laplacian and signless Laplacian matrix of the graph G are defined by and , respectively. For simplifying the nomenclature, we drop the prefix combinatorial to mention . The degree of a graph is denoted by . As well, , for simple graphs. The eigenvalues of Laplacian matrix are . Also, are the eigenvalues of the signless Laplacian matrix. The multi-sets containing and are called -spectra and -spectra respectively. Given any real number we denote and . It is shown in [26], that for any graph and any positive real number .

In quantum mechanics and information theory, a quantum state is represented by a density matrix, which is positive semidefinite, Hermitian and trace-one matrix. As, and are positive semidefinite and Hermitian there are quantum states represented by the density matrices and , where

 ρL(G)=1dL(G) and ρQ(G)=1dQ(G), (2)

respectively [27, 28]. They are called graph Laplacian quantum states. The von-Neumann entropy is a measure of quantum information in a quantum states. In a symbiotic fashion, we define the von-Neumann entropy of a graph which is the von-Neumann entropy of the corresponding graph Laplacian quantum state. The von-Neumann entropies and their generalizations are functions of the eigenvalues of density matrices. Note that, if is an eigenvalue of then is an eigenvalue of . Similarly, is an eigenvalue of . The spectra of a density matrix is a multi-set of eigenvalues of . Thus, or as well as or , which will be clarified from the context. Denote . Therefore, and , when the Laplacian and signless Laplacian spectra are considered, respectively. Throughout the article, we consider logarithm with respect to the base . Now, we are in position to define a number of entropies of graphs, which are as follows:

Sharma-Mittal entropy: The Sharma-Mittal entropy [29, 30, 31, 32] of a graph is denoted by, , and defined by,

 Hq,r(G)=11−r[(Sq(G))1−r1−q−1], (3)

where and are two real parameters , and .

Rényi entropy: The Rényi entropy is a limiting case of the Sharma-Mittal entropy which is defined by:

 H(R)q(G)=limr→1Hq,r(M)=11−qlog(Sq(G)), (4)

where and .

Tsallis entropy: The Tsallis entropy of a graph is defined by:

 H(T)q(G)=limr→qHq,r(G)=11−q(Sq(G)−1), (5)

where and .

Von-Neumann entropy: The Sharma-Mittal entropy reduces to von-Neumann entropy when both and in equation (3), which is

 H(G)=lim(q,r)→(1,1)Hq,r(G)=limq→1H(T)q(G)=−∑γ∈Λ(G)γlog(γ). (6)

The eigenvalue based network parameters are well-investigated in the literature of complex network, such as the Esterda index [33]. Like all of them, there are non-isomorphic graphs with equal entropies. For instance, consider the following two non-isomorphic graphs [34]:

Note that, the spectrum . Hence, their Sharma-Mittal, Rényi, Tsallis and von-Neumann entropies are equal, when they are calculated based on Laplacian spectra. Also, it can be easily checked that for the following two graphs [35] these entropies are equal when they are calculated based on their signless Laplacian spectra.

Now, we calculate the Sharma-Mittal entropy for a number of graphs having explicit expressions for their -spectra. The Laplacian eigenvalues of cycle graph with vertices are where [36]. Degree of a cycle graph is , Hence, the eigenvalues of are . Now, the equation (3) indicates that the Sharma-Mittal entropy of a cycle graph based on its Laplacian eigenvalue is

 Hq,r(Cn)=11−r⎡⎢⎣(2n)q(1−r)1−q(n−1∑j=0sin2q(πjn))1−r1−q−1⎤⎥⎦. (7)

This expression may be further simplified for integer values of [37]. It also appears in the study of Weyl character formula for groups. Similarly, the Rényi, Tsallis, and von-Neumann entropy of cycle graphs are , , and , respectively. It can be easily verified that these entropies are the limiting cases of equation (7).

The Laplacian eigenvalues of the complete graph with vertices are and with multiplicity . The degree of is . Hence, the eigenvalues of the density matrix are , and with multiplicity . Therefore, the Sharma-Mittal entropy of a complete graph based on is

 Hq,r(Kn)=11−r⎡⎢ ⎢⎣{n−1∑i=1(nn(n−1))q}1−r1−q−1⎤⎥ ⎥⎦=11−r[1(n−1)r−1−1]. (8)

The Rényi, Tsallis, and von-Neumann entropy of the complete graph based on Laplacian eigenvalues are , , and , respectively.

The path graph with vertices has Laplacian eigenvalues where . The degree for path graphs is given by . Thus, the eigenvalues of the density matrix is given by . Now the equation (3) indicates that the Sharma-Mittal entropy of a is

 Hq,r(Pn)=11−r⎡⎢ ⎢ ⎢ ⎢⎣⎛⎜ ⎜⎝n−1∑j=0⎛⎜ ⎜⎝2−cos(πjn)2(n−1)⎞⎟ ⎟⎠q⎞⎟ ⎟⎠1−r1−q−1⎤⎥ ⎥ ⎥ ⎥⎦=11−r⎡⎢ ⎢⎣(12(n−1))q(1−r)1−q(n−1∑j=0(2−cos(πjn))q)1−r1−q−1⎤⎥ ⎥⎦. (9)

The Rényi, Tsallis, and von-Neumann entropy of are given by

 (10)

respectively as a limiting case of the equation 9.

The signless Laplacian eigenvalues [38] of cycle graph with vertices are where . Degree of a cycle graph is , Hence, the eigenvalues of are . Now the equation (3) indicates that the Sharma-Mittal entropy of a cycle graph based on its signless Laplacian eigenvalue is

 Hq,r(Cn)=11−r⎡⎢⎣(2n)q(1−r)1−q(n−1∑j=0cos2q(πjn))1−r1−q−1⎤⎥⎦. (11)

Similarly, the Rényi, Tsallis, and von-Neumann entropy of cycle graphs are respectively given by

 H(R)q(Cn)=11−qlog[n−1∑j=0(2n)qcos2q(πjn)],H(T)q(Cn)=11−q[(2n)q(n−1∑j=0cos2q(πjn))−1], andH(Cn)=−n−1∑j=04cos2(πjn)2nlog⎛⎜ ⎜⎝4cos2(πjn)2n⎞⎟ ⎟⎠. (12)

The signless Laplacian eigenvalues of the complete graph with vertices are and with multiplicity . The degree of is . Hence, the eigenvalues of the density matrix are , and with multiplicity . Therefore, after simplification the Sharma-Mittal entropy of a complete graph based on is

 Hq,r(Kn)=11−r⎡⎢⎣{(n−2n)q(n−1)1−q+(2n)q}1−r1−q−1⎤⎥⎦. (13)

The Rényi, Tsallis, and von-Neumann entropy of the complete graph based on signless Laplacian eigenvalues are respectively given by

 (14)

The path graph with vertices has signless Laplacian eigenvalues where . The degree for path graphs is given by . Thus, the eigenvalues of the density matrix is given by . Now the equation (3) indicates that the Sharma-Mittal entropy of a is

 Hq,r(Pn)==11−r⎡⎢ ⎢⎣(2(n−1))q(1−r)1−q(n−1∑j=0cos2q(πj2n))1−r1−q−1⎤⎥ ⎥⎦. (15)

The Rényi, Tsallis, and von-Neumann entropy of are respectively given by

 H(R)q(Pn)=11−qlog[(2n−1)q(n−1∑j=0cos2q(πj2n))],H(T)q(Pn)=11−q[(2n−1)q(n−1∑j=0cos2q(πj2n))−1], andH(Pn)=−n−1∑j=02cos2(πj2n)n−1log⎛⎜ ⎜⎝2cos2(πj2n)n−1⎞⎟ ⎟⎠. (16)

In addition, we can calculate the Sharma-Mittal entropy for a number of graphs, which are not explicitly done here, such as the complete bipartite graph . Recall that, the -eigenvaluies of are given by and with multiplicity , , and , respectively.

3 Bounds on entropies

There are a number of graphs whose -spectra, or -spectra do not have a known general expression. For them we propose a number of bounds on their corresponding entropy functions. We first construct the bound for Sharma-Mittal entropy. Bounds of other entropies are calculated as its limiting cases. Since for any graph the Sharma-Mittal, Rényi, Tsallis and von-Neumann entropy calculated from is greater than that calculated from the .

3.1 Bounds on entropy based on ρL(G)

In this subsections all the entropies are calculated with respect to the eigenvalues of only.

Lemma 1.

Given any graph

 Hq,r(G)≤11−r⎡⎢ ⎢⎣{n(max(di+dj)d)q}1−r1−q−1⎤⎥ ⎥⎦,

where and are degrees of any two vertices of .

Proof.

For any graph the maximum Laplacian eigenvalue [36]. Therefore,

 11−r⎡⎢⎣(n∑i=1(λid)q)1−r1−q−1⎤⎥⎦≤11−r⎡⎢⎣(n(λnd)q)1−r1−q−1⎤⎥⎦. (17)

Putting in the above expression we get the result. ∎

The lemma suggests that an upper bound of Rényi, Tsallis, and von-Neumann entropy are given by
, , and , respectively.

Corollary 1.

For a regular graph , .

Proof.

A -regular graph has equal degree for all the vertices, which is . Hence, and the degree of the graph is . Putting them in lemma 1 we have the result. ∎

Now, for a regular graph with the upper bound of Rényi, Tsallis, and von-Neumann entropy are , , and , respectively.

Recall that, we have denoted , where is an eigenvalue of . Given any bipartite graph it can be proved that [39],

 SL,q(G)≥⎛⎜ ⎜⎝∑id2im⎞⎟ ⎟⎠q+(n−2)⎛⎜ ⎜⎝tnm∑id2i⎞⎟ ⎟⎠qn−2, (18)

which we use in the lemma below.

Lemma 2.

The Sharma-Mittal entropy of a connected bipartite graph with vertices, edges and spanning trees is bounded below by

 11−r⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣1(2m)q(1−r)1−q⎡⎢ ⎢ ⎢ ⎢⎣⎛⎜ ⎜⎝∑id2im⎞⎟ ⎟⎠q+(n−2)⎛⎜ ⎜⎝tnm∑id2i⎞⎟ ⎟⎠qn−2⎤⎥ ⎥ ⎥ ⎥⎦1−r1−q−1⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦,

where are the degrees of the vertices

Proof.

We know that, the degree of a graph with edges is . Therefore, the equation (3) indicates that

 Hq,r(G)=11−r⎡⎢ ⎢⎣1(2m)q(1−r)1−q(SL,q(G))1−r1−q−1⎤⎥ ⎥⎦. (19)

Now applying equation (18) we find the result. ∎

Similarly, the lower bounds of Rényi and Tsallis of connected bipartite graphs are respectively given by,

 11−qlog⎡⎢ ⎢ ⎢ ⎢⎣1(2m)q⎡⎢ ⎢ ⎢ ⎢⎣⎛⎜ ⎜⎝∑id2im⎞⎟ ⎟⎠q+(n−2)⎛⎜ ⎜⎝tnm∑id2i⎞⎟ ⎟⎠qn−2⎤⎥ ⎥ ⎥ ⎥⎦⎤⎥ ⎥ ⎥ ⎥⎦,and11−q⎡⎢ ⎢ ⎢ ⎢⎣1(2m)q⎡⎢ ⎢ ⎢ ⎢⎣⎛⎜ ⎜⎝∑id2im⎞⎟ ⎟⎠q+(n−2)⎛⎜ ⎜⎝tnm∑id2i⎞⎟ ⎟⎠qn−2⎤⎥ ⎥ ⎥ ⎥⎦−1⎤⎥ ⎥ ⎥ ⎥⎦, (20)

as a limiting case of the bound on the Sharma-Mittal entropy.

3.2 Bounds on entropy based on ρQ(G)

In this subsections all the entropies are calculated with respect to the eigenvalues of only.

Lemma 3.

The Sharma-Mittal entropy of a graph

 Hq,r(G)≤11−r⎡⎢ ⎢⎣⎧⎨⎩n(√4m+2(n−1)(n−2)2m)q⎫⎬⎭1−r1−q−1⎤⎥ ⎥⎦.
Proof.

If the graph has edges then degree of the graph . The maximum eigenvalue of is . Therefore,

 11−r⎡⎢⎣{n∑i=1(μid)q}1−r1−q−1⎤⎥⎦≤11−r⎡⎢⎣{n(μn2m)q}1−r1−q−1⎤⎥⎦. (21)

Also, the maximum signless Laplacian eigenvalue [38]. Putting it in the above equation, we find the result. ∎

The above lemma indicates that upper bound of Rényi, Tsallis, and von-Neumann entropy of a graph are respectively given by , , and
.

There are bounds on the largest eigenvalue of in terms of other graph parameters. Let the maximum degree of vertices in be , and the clique number of be . It can be proved that [40].

Lemma 4.

The Sharma-Mittal entropy of a graph with maximum degree and clique number is bounded above by

 11−r⎡⎢⎣[n(2m)q(δ+n(1−1w))q]1−r1−q−1⎤⎥⎦.
Proof.

Note that,

 SQ,q(G)≤n(qn2m)q=n(2m)q(δ+n(1−1w))q. (22)

Putting it in the equation (3) we find the result. ∎

In terms of maximum degree and clique number, the upper bounds of Rényi, Tsallis and von-Neumann entropy of a graph are , , and , respectively.

A lower bound on the Sharma-Mittal entropy of can be constructed in terms of the number of its spanning subgraphs . Let be the number of connected components in . Then the least eigenvalue of [41]

 μ1≥(n−12m)n−1∑S4nc(S), (23)

where the sum is taken over all spanning subgraphs of . It provides the following bound on the Sharma-Mittal entropy of a graph.

Lemma 5.

The Sharma-Mittal entropy of a graph is bounded below by

 11−r⎡⎢ ⎢⎣[n(2m)q[(n−12m)n−1∑S4nc(S)]q]1−r1−q−1⎤⎥ ⎥⎦.

The lower bounds of Rényi, Tsallis and von-Neumann entropy are ,
, and , respectively.

4 Entropy of Product Graphs

Network modelling using product graphs [42] is an interesting method for generating complex networks which may capture the properties of real world networks. Consider two graphs and with and vertices, respectively. The -eigenvalues of and are and , respectively. Similarly the signless-Laplacian eigenvalues of are are given by and , respectively. The produch graph of and is denoted by . A number of graph products has been studies in literature, for instance, the Kronecker product [43], the corona product [44], and many others. We calculate Sharma-Mittal entropy for some of them.

Cartesian product: The Cartesian product of two graphs and is a graph with vertex set where two vertices and are adjacent in if either and or and , where represents the adjacency relation of the respective vertices [44].

Lemma 6.

The Sharma-Mittal entropy of the Cartesian product is given by,

 Hq,r(G)=11−r⎡⎢⎣(1d)q(1−r)1−q(n1∑i=1n2∑j=1(λ(1)i+λ(2)j)q)1−r1−q−1⎤⎥⎦.
Proof.

The eigenvalues of are for [44]. If is the degree of then Shrama-Mittal entropy is given by

 Hq,r(G)=11−r⎡⎢ ⎢ ⎢⎣⎛⎜⎝n1∑i=1n2∑j=1(λ(1)i+λ(2)j)dq⎞⎟⎠1−r1−q−1⎤⎥ ⎥ ⎥⎦=11−r⎡⎢⎣(1d)q(1−r)1−q(n1∑i=1n2∑j=1(λ(1)i+λ(2)j)q)1−r1−q−1⎤⎥⎦. (24)

As the limiting cases of the Sharma-Mittal entropy the Rényi, Tsallis, and von-Neumann entropy of the product graph are respectively given by

 (25)

Kronecker product: The Kronecker product of two graphs and is a graph with vertex set where two vertices and are adjacent in if either and [44].

Lemma 7.

Let and be connected regular graphs with regularities and , respectively. Then the Sharma-Mittal entropy of is given by,

 Hq,r(G)=11−r⎡⎢⎣(1d)q(1−r1−q(n1∑i=1n2∑j=1(Kλ(2)j+λ(1)iS−λ(1)iλ(2)j)q)1−r1−q−1⎤⎥⎦.
Proof.

The eigenvalues of are for and [44]. If is the degree of then Sharma-Mittal entropy is given by

 Hq,r(G)=11−r⎡⎢ ⎢ ⎢⎣⎛⎜⎝n1∑i=1n2∑j=1⎛⎜⎝Kλ(2)j+λ(1)iS−λ(1)iλ(2)jd⎞⎟⎠q⎞⎟⎠1−r1−q−1⎤⎥ ⎥ ⎥⎦=11−r⎡⎢⎣(1d)q(1−r1−q(n1∑i=1n2∑j=1(Kλ(2)j+λ(1)iS−λ(1)iλ(2)j)q)1−r1−q−1⎤⎥⎦. (26)

The Rényi entropy, Tsallis entropy and von-Neumann entropy of are

 (27)

Strong Product: The Strong Product of two graphs and is a graph with vertex set where two vertices and are adjacent if either and in ; or in and ; or in and in [44].

Lemma 8.

Let and be connected regular graphs with regularities and , respectively. Then the Sharma-Mittal entropy is given by,