# Entropy of Weighted Graphs with Randi´c Weights

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

_{i}, let d

_{i}be the degree of v

_{i}. For an edge v

_{i}v

_{j}, one defines:

_{i}v

_{j}) is the weight (or volume) of the edge v

_{i}v

_{j}and w(v

_{i}v

_{j}) > 0. The node entropy has been defined by:

**Definition 1.**For an edge weighted graph G = (V, E, w), the entropy of G is defined by:

_{n}, P

_{n}and S

_{n}be the complete graph, the path graph and the star graph with n vertices, respectively. A tree is called a subdivided star if it is obtained from a star by subdividing each edge of the star exactly once, and at most one edge is subdivided twice. The double star with n vertices, denoted by S

_{p,q}, is the tree obtained by connecting two centers of two stars S

_{p}and S

_{q}, where p + q = n. The balanced complete bipartite graph is a complete bipartite graph, such that the numbers of vertices in the two parts are equal or have a difference of one. The balanced complete multipartite graph is a complete multipartite graph, such that the number of vertices in any two parts are equal or have a difference of one, which is also called the Turán graph.

## 3. Extremal Properties of I(G, w)

**Theorem 1.**Let G = (V, E, w) be a graph with n vertices. If w(e) = c for each edge e, where c > 0 is a constant, then we obtain:

**Proof.**Suppose m = |E|. Since w(e) = c for each edge e, then we get:

^{α}, where d(u) denotes the degree of u. Then, the general Randić index [44,52,53] is defined as:

_{α}(G) for α < 0 that are used in this paper.

**Lemma 1**([44]). (i) Let G be a graph with n vertices and no isolated vertices. For α ∈ (−1/2, 0), the maximum value of R

_{α}is$\frac{n{(n-1)}^{1+2\alpha}}{2}$ and the minimum value is min $\mathrm{min}\{{(n-1)}^{1+\alpha},\frac{n}{2}(even\phantom{\rule{0.2em}{0ex}}n),\frac{n-3}{2}+{2}^{1+\alpha}(odd\phantom{\rule{0.2em}{0ex}}n)\}$; for α ∈ (−∞, −1), the maximum value of R

_{α}is$\frac{n}{2}(even\phantom{\rule{0.2em}{0ex}}n)\phantom{\rule{0.2em}{0ex}}or\phantom{\rule{0.2em}{0ex}}\frac{n-3}{2}+{2}^{1+\alpha}(odd\phantom{\rule{0.2em}{0ex}}n)$, and the minimum value is$\frac{n{(n-1)}^{1+2\alpha}}{2}$.

_{n}attains the minimum value of R

_{α}for α < 0 and R

_{α}(S

_{n}) = (n − 1)

^{α}

^{+1}; the path graph P

_{n}attains the maximum value of R

_{α}for α ∈ [−1/2, 0] and R

_{α}(P

_{n}) = 2

^{α}

^{+1}+ (n − 3)4

^{α}; the subdivided star attains the maximum value of R

_{α}for α ∈ [−∞, −2] when n ≥ 7, and the R

_{α}-value of the subdivided star is$\frac{n-1}{2}({(n-1)}^{\alpha}+{2}^{\alpha})\phantom{\rule{0.2em}{0ex}}(odd\phantom{\rule{0.2em}{0ex}}n)\phantom{\rule{0.2em}{0ex}}or\frac{n-2}{2}({(n-2)}^{\alpha}+{2}^{\alpha})+{4}^{\alpha}(even\phantom{\rule{0.2em}{0ex}}n)$.

_{1}, d

_{2},…, d

_{n}) denote the adjacency matrix of G and the diagonal matrix of vertex degrees, respectively. Let λ

_{1}(G) ≥ λ

_{2}(G) ≥ … ≥ λ

_{n}(G) = 0 be the eigenvalues of L(G), which are also called Laplacian eigenvalues of G. In [59], the authors proved the following result.

**Lemma 2**([59]). Let G be a simple connected graph with n vertices. Then:

_{α}.

**Theorem 2.**Let G be a connected graphs with n vertices. For α < 0, we have:

_{α}(G) and R = max R

_{α}(G).

**Proof.**Since α < 0, we have:

_{α}for α < 0, we can obtain an upper bound or a lower bound of I(G, α). As an example, we can get some bounds of R

_{α}from Lemmas 1 and 2.

**Corollary 1.**(i) Let G be a graph with n vertices and no isolated vertices. For α ∈ (−1/2, 0), we have:

**Corollary 2.**Let G be a simple connected graph with n vertices. Then:

**Corollary 3.**Let G be a graph with n vertices. Let δ and Δ be the minimum degree and the maximum degree of G, respectively. Then, for α < 0, we have:

_{α}(G) and R = max R

_{α}(G).

**Theorem 3.**Let G = (V, E, w) be a regular graph with n vertices and n ≥ 3. Then, we have:

**Proof.**Suppose G = (V, E, w) is k-regular. Then, k ≥ 2, since G is connected and n ≥ 3. Therefore, we have:

**Theorem 4.**Let G = (V, E, w) be a complete bipartite graph with n vertices. Then, we infer:

**Proof.**Suppose G = (V, E, w) is a complete bipartite graph with n vertices, and the two parts have p and q vertices, respectively. Therefore, p + q = n. We have:

_{n−t}of which one end vertex coincides with a pendent vertex of a star S

_{t}

_{+1}of order t+1. Observe that CS(n, t) is the path graph if t = 2 and is the star graph if t = n − 1. Then, for 2 ≤ t ≤ n − 2, we have:

**Theorem 5.**Among all comets with n vertices and parameter t,

- for α = 1, we have:$$I(CS(n,{t}_{0}),\alpha )\le I(CS(n,t),\alpha )\le \phantom{\rule{0.2em}{0ex}}\mathrm{log}(n-1),$$
_{0}, where t ≥ 3 is the root the equation$\frac{\partial I(CS(n,t),1)}{\partial t}=0,$ i.e.,$$(({t}^{2}+t)\mathrm{log}\phantom{\rule{0.2em}{0ex}}t-6t+8n-14)(2t-3)=({t}^{2}-3t+4n-6)((2t+1)\mathrm{log}\phantom{\rule{0.2em}{0ex}}t-\frac{t-4}{\mathrm{ln}\phantom{\rule{0.2em}{0ex}}2}-6).$$ - For α = −1, we have:$$I(CS(n,t),\alpha )\le \phantom{\rule{0.2em}{0ex}}\mathrm{log}(n-1),$$
_{0}, where t^{0}≥ 4 is the root the equation$\frac{\partial I(CS(n,t),-1)}{\partial t}=0,$ i.e.,$$(1+(2t-1)\mathrm{log}\phantom{\rule{0.2em}{0ex}}t+(n-t)t)(2-{t}^{2})=\left(2t-1+\frac{(n-t)t}{2}\right)\left(-2-2{t}^{2}+2\phantom{\rule{0.2em}{0ex}}\mathrm{log}\phantom{\rule{0.2em}{0ex}}t+\frac{4t-{t}^{2}}{\mathrm{ln}2}\right).$$

_{0}and t

^{0}

_{0}as follows in Table 1.

**Theorem 6.**For S

_{p,q}, we have that for α ∈ [0.5, +∞),

**Theorem 7.**Let D(t, r) be a dendrimer with n vertices with only one center. Then, for α ∈ (−∞, 0), we have:

**Proof.**If r = 1, i.e., D is a star, then we have I (D, α) = log (t + 1). Since D (t, r) has only one center, we have $n=1+\frac{(t+1)({t}^{r}-1)}{t-1}=t+2$, i.e., t= n − 2. Therefore, in this case, we have I (D, α) = log (n−1). If t = 1, i.e., D is a path, then by some elementary calculations, we have:

^{r−}

^{1}leaves, and both end vertices of any other edge have degree t + 1. Set ${A}_{1}={\displaystyle {\sum}_{uv\in E}\frac{{(d(u)d(v))}^{\alpha}}{{\displaystyle {\sum}_{uv\in E}{(d(u)d(v))}^{\alpha}}}}$. Then, we infer:

## 4. Summary and Conclusions

**Problem 1.**Determine extremal values of I(T, α) among all trees with n vertices for any real number α.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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n | 30 | 40 | 50 | 60 | 100 | 200 | 300 | 400 | 500 | 1000 | |
---|---|---|---|---|---|---|---|---|---|---|---|

t_{0} | 11 | 15 | 19 | 22 | 25 | 36 | 57 | 74 | 88 | 102 | 155 |

${t}_{0}^{{}^{\prime}}$ | 18 | 27 | 36 | 45 | 54 | 92 | 190 | 288 | 387 | 486 | 983 |

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**MDPI and ACS Style**

Chen, Z.; Dehmer, M.; Emmert-Streib, F.; Shi, Y.
Entropy of Weighted Graphs with Randi´c Weights. *Entropy* **2015**, *17*, 3710-3723.
https://doi.org/10.3390/e17063710

**AMA Style**

Chen Z, Dehmer M, Emmert-Streib F, Shi Y.
Entropy of Weighted Graphs with Randi´c Weights. *Entropy*. 2015; 17(6):3710-3723.
https://doi.org/10.3390/e17063710

**Chicago/Turabian Style**

Chen, Zengqiang, Matthias Dehmer, Frank Emmert-Streib, and Yongtang Shi.
2015. "Entropy of Weighted Graphs with Randi´c Weights" *Entropy* 17, no. 6: 3710-3723.
https://doi.org/10.3390/e17063710