# Graphs with minimum degree-based entropy

The degree-based entropy of a graph is defined as the Shannon entropy based on the information functional that associates the vertices of the graph with the corresponding degrees. In this paper, we study extremal problems of finding the graphs attaining the minimum degree-based graph entropy among graphs and bipartite graphs with a given number of vertices and edges. We characterize the unique extremal graph achieving the minimum value among graphs with a given number of vertices and edges and present a lower bound for the degree-based entropy of bipartite graphs and characterize all the extremal graphs which achieve the lower bound. This implies the known result due to Cao et al. (2014) that the star attains the minimum value of the degree-based entropy among trees with a given number of vertices.

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## 1 Introduction

All graphs considered in this paper are simple, finite and undirected. The logarithms here are base . We use the convention that .

A system that consists of a large number of microscopic components interacting with each other appears to be complex always [16]. In order to describe the complexity or the information of a system, various graph entropies were introduced (refer to [3, 7, 8] for reviewing). Reshevsky is the pioneer to quantify the complexity of a system by the so-called topological information content, which is the earliest graph entropy measurement [15]. Since then, many graph entropies based on various graph invariants, such as the number of matchings, independent sets and spanning forests, and the Randić index [4, 5, 17, 18], have been studied. Compared with other entropies, Cao et al. [5] proposed an easily computable graph entropy called degree-based entropy, which has been proved useful in information theory, complexity networks and mathematical chemistry [1, 2].

Let be a graph vertex set , and let be the degree of of . The degree-based graph entropy of is defined as

 Id(G) =−n∑i=1dG(vi)∑nj=1dG(vj)logdG(vi)∑nj=1dG(vj). (1)

One of the attracting and challeging problem in the theoretic study of graph entropy is giving extremal values of entropies and determing the correspoding graphs. Cao et al. [5] proved some extremal values for the degree-based entropy of some certain families of graphs, such as trees, unicyclic graphs and bicyclic graphs, and proposed conjectures to determine extremal values. Shortly afterwards Ilić [12] proved one part of the conjectures. Ghalavand et al. [10] applied majorization technique to extend some known results on the extremal values of the degree-based entropy of trees, unicyclic and bicyclic graphs. Dong et al. [9] obtained the maximum value of the degree-based entropy among bipartite graphs with a given number of vertices and edges by characterizing corresponding degree sequences. Motivated by these results on extremal values of degree-based entropy, we study the problem of minimum values of the degree-based entropy among graphs and bipartite graphs with a given number of vertices and edges.

A graph (resp. bipartite graph) with vertices and edges is referred to as an -graph (resp. -bipartite graph). A simple graph in which each pair of distinct vertices is joined by an edge is called a complete graph. Up to isomorphism, there is just one complete graph with vertices, denoted by . A complete bipartite graph is a simple bipartite graph with bipartition in which each vertex of is joined to each vertex of ; if and , such a graph is denoted by . Let be a simple graph. The complement of is the simple graph whose vertex set is and whose two distinct vertices are adjacent if and only if they are not adjacent in . The union of simple graphs and , denoted by , is the graph with the vertex set and the edge set . Let and be two integers, and let and . Note that the size of a maximum clique of a graph with edges cannot be more than . The graph is obtained by adding a vertex adjacent to vertices of . Let

 σ(x)={0,ifx=0;1,otherwise.

We characterize the extremal graph achieving the minimum value among -graphs.

###### Theorem 1.

Let be an -graph with , . Then

 Id(G)≥log(2m)−t∗k∗logk∗+(k∗−t∗)(k∗−1)log(k∗−1)+t∗logt∗2m,

the equality holds if and only if

We will use the following theorem that characterizes all the extremal graphs achieving a lower bound among bipartite graphs with a given number of vertices and edges to prove the above theorem.

###### Theorem 2.

Let be an -bipartite graph with and . Then the equality holds in the inequality if and only if , where and satisfy and .

The following result due to Cao et al. [5] is immediately implied by Theorem 2.

###### Corollary 1 ([5]).

Let be a tree on vertices. Then , the equality holds in the inequality if and only if .

The rest of this paper is organized as follows. In Section 2, we give some preliminary results that will be used later. In Section 3, we give the proofs of our main results. We conclude this paper with a problem and some remarks.

## 2 Preliminaries

Let be a graph, and let be the number of vertices with degree in a set for . Denote by the degree sequence of in which and . By denote the degree sequence of . Every graph with the degree sequence is a realization of . A degree sequence is unigraphic if all its realizations are isomorphic. We use and to denote the maximum degree of vertices in a set of vertices and a graph , respectively. The size of a maximum clique in is denoted by .

Let be a graph with edges, and let be the vertex set of . By the hand shaking lemma,

 n∑j=1dG(vj)=2m.

From Equality (1), we infer

 Id(G) =logn∑j=1dG(vj)−1∑nj=1dG(vj)n∑i=1dG(vi)logdG(vi) =log(2m)−12mn∑i=1dG(vi)logdG(vi).

We define a function

 hd(G)=n∑i=1dG(vi)logdG(vi).

Equation (1) implies that for a certain family of graphs .

Let us introduce the concept of majorization as a useful tool to solve some inequality problems [14]. Let and be integer sequences of length . We define that majorizes , denoted by , if

 s∑i=1ai≥s∑i=1bi,  s=1,2,…,n−1

and

 n∑i=1ai=n∑i=1bi.

The majorization is strict, and is denoted by , if at least one of the inequalities is strict. Let be integers satisfying .

Ghalavand et al. [10] extended some known results about the minimum and maximum values of the degree-based entropy by the following theorem.

###### Theorem 3 ([10]).

Let and be two -graphs. If , then , the equality holds in this inequality if and only if .

Threshold graphs have been defined by Chvátal and Hammer [6] as follows. A graph is called a threshold graph if there exist non-negative real numbers associated with the vertex set of and such that for any two distinct vertices and , and are adjacent if and only if . Let be a collection of sets. The collection is nested if for every two sets in , one is a subset of the other. Some basic characterizations of threshold graphs are given by the following theorem.

###### Theorem 4 ([6]).

Let be a graph. Then the following are equivalent:

(a)

the graph is a threshold graph;

(b)

there are no four distinct vertices such that and ;

(c)

the graph is a split graph (i.e., its vertices can be partitioned into a clique and a stable set ) and the neighborhoods of the vertices of are nested.

The family of difference graphs was introduced by Hammer et al. [11]. A graph is said to be a difference graph if there exist real numbers associated with the vertex set of and a positive real number such that

(a)

for ;

(b)

two distinct vertices and are adjacent if and only if .

The following result is due to Mahadev and Peled [13].

###### Theorem 5 ([13]).

Let be a bipartite graph with bipartition . Then the following conditions are equivalent:

(a)

the graph is a difference graph;

(b)

there are no and such that and ;

(c)

every induced subgraph without isolated vertices has on each side of the bipartition a domination vertex, that is, a vertex is adjacent to all the vertices on the other side of the bipartition.

The conjugate sequence of a sequence is the sequence in which . The following result is one characterization of difference graphs by the degree sequences.

###### Theorem 6 ([13]).

A pair of non-negative and non-increasing integer sequences and with is a pair of degree sequences of a difference graph with bipartition if and only if , where is the conjugate of .

We need the next lemma to prove Theorem 1.

###### Lemma 1.

Let be an -graph with and . If attains the minimum value among -graphs, then is a threshold graph.

###### Proof.

Suppose that is not a threshold graph. By Theorem 4 (b), there are four vertices such that and . We may assume without loss of generality that . Set . Hence . By Theorem 3, we have , a contradiction. ∎

To prove Theorem 2, we need the next lemma.

###### Lemma 2.

Let be an -bipartite graph with and . If attains the minimum value among -bipartite graphs, then is a difference graph.

###### Proof.

Let be the bipartition of . Suppose that is not a difference graph. By Theorem 5(b), there are four vertices and such that and . We may assume without loss of generality that . Set . Hence . By Theorem 3, we have , a contradiction. ∎

## 3 Proofs

###### Proof of Theorem 1.

Suppose that is an -graph such that attains the minimum value among graphs with vertices and edges. It follows from Lemma 1 that is a threshold graph. And by Theorem 4 (c), the graph is a split graph (i.e., its vertices can be partitioned into a clique and a stable set ) and the neighborhoods of the vertices of are nested. If is not the maximum clique, then there exists a vertex in the maximum clique. Since is the stable set, there is at most one vertex in the maximum clique. This implies that vertices of can be partitioned into a clique and a stable set . So we may assume is the maximum clique. Let and . Then we have . It follows from that . Since , it is easy to get that . Thus .

Denote by be the set of neighbors of vertices in , and the subgraph of induced by all edges between and . So is a bipartite graph. Let be the bipartition of in which and . If (i.e., and ), then . So we consider (i.e., ) in the following. For and , we analyze conditional extremums of the following function

 f(z1,z2,…,z|X|)=|X|∑j=1(zj+k−1)log(zj+k−1)−|X|∑j=1zjlogzj

and corresponding Lagrangian function

 L(z1,z2,…,z|X|,λ)=f(z1,z2,…,z|X|)+λ⎛⎝|X|∑j=1zj−t⎞⎠

with the additional boundary condition . The function is well-defined and differentiable with respect to each of its arguments on the closed region, so the extremal values can be either critical points or on the boundary. By direct calculation, we obtain

 ∂∂λL=|X|∑j=1zj−t=0

and

 ∂∂zjL=log(zj+k−1)−logzj+λ=0

for . It follows from this set of equations that the unique critical point satisfies . Using the second order conditions, this is a local maximum, as

 ∂2∂z2jL=1(zj+k−1)ln2−1zjln2<0,

where .

For variables satisfying , the function
attains the maximum value if and only if for . And by Theorem 2, we have

 hd(G∗)= hd(Kk)+hd(H)−|X|(k−1)log(k−1) −|X|∑j=1dH(xj)logdH(xj) +|X|∑j=1(dH(xj)+k−1)log(dH(xj)+k−1) ≤ k(k−1)log(k−1)+tlogt−|X|(k−1)log(k−1) −tlogt|X|+|X|(t|X|+k−1)log(t|X|+k−1), (2)

the equality holds in this inequality if and only if for .

Let for . By calculation, we have

 ddxp= (k−1)log(1+tx(k−1))>0.

This implies that is rigidly monotonically increasing for . Let . Since , . Therefore, we have

 k(k−1)log(k−1)+tlogt−|X|(k−1)log(k−1) −tlogt|X|+|X|(t|X|+k−1)log(t|X|+k−1) ≤ k(k−1)log(k−1)+tlogt−t′(k−1)log(k−1) −tlogtt′+t′(tt′+k−1)log(tt′+k−1), (3)

the equality holds in this inequality if and only if .

Let for . By calculation, we have

 ddyg= t′ln(t′(y−1)+tt′)+(2y−t′−1)ln(y−1)+2yln2 > t′ln(t′+tt′) > 0.

This implies that is rigidly monotonically increasing for . So we have

 k(k−1)log(k−1)+tlogt−t′(k−1)log(k−1) −tlogtt′+t′(tt′+k−1)log(tt′+k−1) ≤ k∗(k∗−1)log(k∗−1)+tlogt−t′(k∗−1)log(k∗−1) −tlogtt′+t′(tt′+k∗−1)log(tt′+k∗−1), (4)

the equality holds in this inequality if and only if .

If , then we have

 k∗(k∗−1)log(k∗−1)+tlogt−t′(k∗−1)log(k∗−1) −tlogtt′+t′(tt′+k∗−1)log(tt′+k∗−1) = t′k∗logk∗+(k∗−t′)(k∗−1)log(k∗−1)+t′logt′. (5)

We prove (i.e., ) when . It follows from that . Suppose that for . So we have for . This implies which contradicts . So we have when . It follows from that . We consider . This implies . And by (1), (2), (3), (4) and (5), we have

 Id(G)≥log(2m)−t∗k∗logk∗+(k∗−t∗)(k∗−1)log(k∗−1)+t∗logt∗2m,

the equality holds in this inequality if and only if

 G≅K(k∗,t∗)∪¯¯¯¯¯Kn−k∗−1.

###### Proof of Theorem 2.

Suppose that is an -bipartite graph such that attains the minimum value among bipartite graphs with vertices and edges. Assume that . Let be the bipartition of . We denote the vertices of and by and , respectively. From Lemma 2, is a difference graph. By Theorem 5 (c), without isolated vertices, has a domination vertex on each side of the bipartition. Without loss of generality, we assume that is the domination vertex in and for . So we have . If , then .

For , we prove the extremal graphs by induction on . For , . For , set and assume that .

We have a recursive relation

 hd(G∗)= |X|∑i=1dG∗(xi)logdG∗(xi)+n−|X|∑j=1dG∗(yj)logdG∗(yj) = |X|∑i=1dG∗(xi)logdG∗(xi)+b∑j=1dG∗(yj)logdG∗(yj) = |X|∑i=2dG∗(xi)logdG∗(xi)+b∑j=1(dG∗(yj)−1)log(dG∗(yj)−1)+b∑j=1dG∗(yj)logdG∗(yj) −b∑j=1(dG∗(yj)−1)log(dG∗(yj)−1)+blogb = hd(G′)+b∑j=1dG∗(yj)logdG∗(yj)−b∑j=1(dG∗(yj)−1)log(dG∗(yj)−1)+blogb.

For , we analyze conditional extremums of the following function

 f(z1,z2,…,zb)=b∑j=1zjlogzj−b∑j=1(zj−1)log(zj−1)

and corresponding Lagrangian function

 L(z1,z2,…,zb,λ)=f(z1,z2,…,zb)+λ(b∑j=1zj−m)

with the additional boundary condition . The function is well-defined and differentiable with respect to each of its arguments on the closed region, so the extremal values can be either critical points or on the boundary. By direct calculation, we obtain

 ∂∂λL=b∑j=1zj−m=0

and

 ∂∂zjL=logzj−log(zj−1)+λ=0

for . It follows from this set of equations that the unique critical point satisfies . Furthermore, using the second order conditions, it follows that this is a local maximum, as

 ∂2∂z2jL=−1ln2zj(zj−1)<0.

For variables , the function attains the maximum value if and only if for . So we have in which . By Theorem 6, we have , where is the conjugate of . This implies that the degree sequence of is . Obviously, this pair of degree sequences is unigraphic, that is, is the only realization of this pair of degree sequences up to isomorphism. If , then there exist two integers and such that and . By induction hypothesis, for . By calculation, we have

 Id(Kq,b∪¯¯¯¯¯Kn−q−b)= log(2m)−12m(bqlogq+qblogb) = log(2m)−12m(mlogm) = 1+log√m.

Therefore we have , the equality holds in the inequality if and only if , where and satisfy and .

Now we consider in the following. It is sufficient to prove if does not hold for any integers and . Contrarily, we assume that . It follows that . So we have . This implies attains the minimum value among -bipartite graphs and is not isomorphic to for any integers and , which contradicts that for . ∎

## 4 Concluding remarks

Among bipartite graphs with vertices and edges, we prove that achieves the lower bound , where and . As an example, we consider the extremal results on bipartite with at most six vertices. Before we list the results, we define a special bipartite graph. Let , and be integers with and . Set , . Let be the -bipartite graph with bipartition , where , , , , for , and for . The bipartite graphs attaining the minimum value of degree-based entropy among