First and Second Maximum of Randi\'{c} Index Among all $k-$Cyclic Graphs of a Given Order

TL;DR

This paper determines the graphs with the highest and second-highest Randić index among all k-cyclic graphs with a fixed number of vertices, advancing understanding of extremal properties in graph theory.

Contribution

It explicitly computes the first and second maximum Randić indices among all n-vertex k-cyclic graphs, filling a gap in extremal graph theory.

Findings

Identified the graphs with maximum Randić index for given n and k.

Determined the second maximum Randić index for these graphs.

Abstract

Suppose $G$ is a simple graph with edge set $E(G)$. The Randi\'{c} index $R(G)$ is defined as $R(G)=\sum_{uv\in E(G)}\frac{1}{\sqrt{deg_{G}(u)deg_{G}(v)}}$, where $deg_G(u)$ denotes the vertex degree of $u$ in $G$. In this paper, the first and second maximum of Randi\'{c} index among all $n-$vertex $k-$cyclic graphs were computed.

Equations

$$R(G_2)-R(G_1)$$

$$R(G^{\prime })-R(G)$$

$${\Omega }_1(n)$$

$${\Omega }_2(n)$$

$${\Omega }_3(n)$$

$${\Omega }_4(n)$$

$${\Lambda }_k^1(n)$$

$${\Gamma }_k^1(n)$$

$${\Omega }_5(n)$$

$${\Omega }_6(n)$$

$${\Omega }_7(n)$$

$${\Omega }_8(n)$$

$${\Lambda }_k^2(n)$$

$${\Gamma }_k^2(n)$$

Full text

Keywords: $k$-Cyclic graph, Randić index, graph transformation.

2010 AMS Subject Classification Number: $05C15$.

1 Definitions and Notations

In this section, we first describe some mathematical notions that will be kept throughout. All graphs considered in this paper are assumed to be simple, undirected and finite without multiple edges. The undefined terms and notations are from [6, 7].

The degree of a vertex $v$ in $G$ is denoted by $deg_{G}{(v)}$ and $N[v,G]$ is the set of all vertices adjacent to $v$. The notations $\Delta=\Delta(G)$ and $n_{i}=n_{i}(G)$ are used for the maximum degree and the number of vertices of degree $i$ in $G$, respectively. The number of edges connecting a vertex of degree $i$ with a vertex of degree $j$ in $G$ is denoted by $m_{i,j}(G)$. An connected $n-$vertex graph $G$ is called to be $c$-cyclic if it has $n+c-1$ edges. The number $c=c(G)$ is said to be the cyclomatic number of $G$.

Suppose $W$ is a non-empty subset of vertices in a graph $G$. The subgraph of $G$ obtained by deleting the vertices of $W$ is denoted by $G-W$ and similarly, if $F\subseteq E(G)$, then the subgraph obtained by deleting all edges in $F$ is denoted by $G-F$. In the case that $W=\{v\}$ or $F=\{xy\}$, the subgraphs $G-W$ and $G-F$ will shortly be written as $G-v$ or $G-xy$, respectively. Furthermore, if $x$ and $y$ are nonadjacent vertices in $G$, then the notation $G+xy$ is used for the graph obtained from $G$ by adding an edge $xy$.

The Randić index of a graph $G$ is defined as $R(G)=\sum_{uv\in E(G)}\frac{1}{\sqrt{deg_{G}(u)deg_{G}(v)}}$ [9]. The most important mathematical properties of this number were presented in [6, 7]. In the following, we first briefly review the literature on ordering graphs with Randić index.

In [1, 2], the first and second maximum of Randić index in the class of all $n-$vertex $c$-cyclic graphs, $c=3,4$, were obtained. Shiu and Zhang [11] obtained the maximum value of Randić index in the class of all $n-$vertex chemical trees with $k$ pendents such that $n<3k-2$. Shi [10] obtained some interesting results for chemical trees with respect to two generalizations of Randić index. For related results we refer to the survey article of Li and Shi [8] on the topic of Randić index.

Deng et al. [3] considered various degree mean rates of an edge and gave some tight bounds for the variation of the Randić index of a graph $G$ in terms of its maximum and minimum degree mean rates over its edges.

2 Five Graph Transformations

In this section five graph transformations will be presented which are useful in computing Randić index of graphs. The Transformations I and II were introduced in [5].

Transformation I. Suppose that $G$ is a graph with a given vertex $w$ such that $deg_{G}(w)\geq 1$. In addition, we assume that $P:=v_{1}v_{2}\ldots v_{k}$ and $Q:=u_{1}u_{2}\ldots u_{l}$ are two paths of lengths $k$ and $l$, respectively. Let $G_{1}$ be the graph obtained from $G$, $P$ and $Q$ by attaching edges $v_{1}w$ and $wu_{1}$. Define $G_{2}=G_{1}-v_{1}w+u_{k}v_{1}$. 2. 2.

Transformation II. Suppose that $G$ is a graph with given vertices $x$ and $y$ such that $deg_{G}(x),deg_{G}(y)\geq 2$ and for all $v\in N[x,G]$, $deg_{G}(x)\geq 2$. In addition, we assume that $P:=v_{1}v_{2}...v_{l}$ and $Q:=u_{1}u_{2}...u_{k}$ are two paths of lengths $l$ and $k$, respectively. Define $G_{1}$ to be the graph obtained from $G$, $P$ and $Q$ by attaching vertices $yv_{1}$, $u_{1}x$, and $G_{2}=G_{1}-u_{1}x+v_{l}u_{1}$. 3. 3.

Transformation III. Suppose that $G$ is a graph with vertices $x,y,w$ and $z$ such that $\{xy,wz\}\subseteq E(G)$. In addition, we assume that $G^{\prime}$ is a trivial graph with vertex set $\{v\}$. Define $G_{1}=G-\{xy\}+\{xv,vy\}$ and $G_{2}=G-\{wz\}+\{wv,vz\}$. 4. 4.

Transformation IV. Suppose that $G$ is a graph with vertices $v_{1},v_{2},v_{3},v_{4}$ and $u_{1}$ such that $\{v_{1}v_{2},v_{2}v_{3},v_{4}u_{1}\}\subseteq E(G)$, $deg_{G}(v_{1})\geq 2$, $deg_{G}(v_{2})=2$, $deg_{G}(v_{3})\geq 3$ and $deg_{G}(v_{4})\geq 4$. Define $G^{\prime}=G-\{v_{4}u_{1}\}+\{v_{2}u_{1}\}$. 5. 5.

Transformation V. Suppose that $G$ is a graph with vertices $x_{1},x_{2},x_{3},x_{4},x_{5},x_{6}$ and $w$ such that $\{x_{1}x_{2},x_{2}x_{3},x_{2}w,x_{4}x_{5},x_{5}x_{6}\}\subseteq E(G)$, $deg_{G}(x_{1})=deg_{G}(x_{2})=deg_{G}(x_{3})=3$, $deg_{G}(x_{4})=4$, $deg_{G}(x_{5})=2$ and $deg_{G}(x_{6})=1$ or $2$. Define $G^{\prime}=G-\{x_{2}w\}+\{x_{5}w\}$.

It is well-known that if the derivative $f^{\prime}(x)$ of a continuous function $f(x)$ satisfies $f^{\prime}(x)>0$ on an open interval $(a,b)$, then $f(x)$ is increasing on $(a,b)$.

Lemma 2.1**.**

The following hold:

Let $G_{1}$ and $G_{2}$ be two graphs satisfying the conditions of Transformation $I$. Then $R(G_{2})>R(G_{1})$. 2. 2.

Let $G_{1}$ and $G_{2}$ be two graphs satisfying the conditions of Transformation $II$. Then $R(G_{2})>R(G_{1})$. 3. 3.

Let $G_{1}$ and $G_{2}$ be two graphs as shown in Transformation $III$.

1. (a)

If $deg_{G}(x),deg_{G}(y)\geq 3$ and $deg_{G}(z)\in\{1,2\}$, then $R(G_{2})>R(G_{1})$. 2. (b)

If $deg_{G}(x)=2$, $deg_{G}(w)\geq 3$ and $deg_{G}(z)=1$, then $R(G_{2})>R(G_{1})$. 3. (c)

If $deg_{G}(x)=2$ and $deg_{G}(w)=2$, then $R(G_{2})=R(G_{1})$. 4. 4.

Let $G$ and $G^{\prime}$ be two graphs satisfying the conditions of Transformation $IV$, $\Delta(G)=q$, $N[v_{4},G]=\{u_{1},\ldots,u_{q}\}$ and $deg_{G}(u_{i})=d_{i}$, $1\leq i\leq q$. If $d_{1},d_{2}\leq 3$, then $R(G^{\prime})\geq R(G)$, with equality if and only if $deg_{G}(v_{1})=2$, $deg_{G}(v_{3})=d_{1}=d_{2}=3$ and $q=d_{3}=d_{4}=4$. 5. 5.

Let $G$ and $G^{\prime}$ be two graphs satisfying the conditions of Transformation $V$. Then $R(G^{\prime})<R(G)$.

Proof.

Let $deg_{G}(w)=x$ and $k,l\geq 2$. Then by the proof of [5, Lemma 3.1],

[TABLE]

Let $f(x):=\frac{1}{\sqrt{2(x+1)}}-\frac{2}{\sqrt{2(x+2)}}$, for $x\in(0.8,\infty)$. Since $f(x)$ is continuous on the open interval $(0.8,\infty)$ and $f^{\prime}(x)>0$ on this interval, $f$ is increasing on $(0.8,\infty)$. Therefore, by Equation 2.1, $R(G_{2})-R(G_{1})>0.01$. The proof of other cases of $k$ and $l$ are similar and we omit them. 2. 2.

Let $deg_{G}(x)=s$ and $k,l\geq 2$. Then by the proof of [5, Lemma 3.4],

[TABLE]

Let $g(x):=\sqrt{x}-\sqrt{x+1}$, for $x\in(0,\infty)$. Then $g$ is increasing on $(0,\infty)$ and so by Equation 2.2, $R(G_{2})-R(G_{1})\geq 6\times 10^{-10}$. The proof of other cases of $k$ and $l$ are similar and we omit them. 3. 3.

Suppose $deg_{G}(x)=s$, $deg_{G}(y)=r$, $deg_{G}(w)=l$ and $deg_{G}(z)=t$. To prove $(a)$, we note that

[TABLE]

To prove $(b)$, we first calculate the difference between $R(G_{2})$ and $R(G_{1})$.

[TABLE]

Let $h(x)=\frac{1}{\sqrt{2x}}-\frac{1}{\sqrt{x}}$, for $x\in(0,\infty)$. Then again $h$ is increasing on $(0,\infty)$ and hence by Equation 2.3, $R(G_{2})-R(G_{1})>0.038$. For the proof of $(c)$, it is enough to notice that $m_{i,j}(G_{1})=m_{i,j}(G_{2})$, $1\leq i\leq j\leq n-1$. Thus $R(G_{2})=R(G_{1})$, as desired. 4. 4.

Suppose that $deg_{G}(v_{1})=s$, $deg_{G}(v_{3})=r$ and $deg_{G}(v_{4})=q$. Then,

[TABLE]

Let $f(x):=\frac{1}{\sqrt{ax}}-\frac{1}{\sqrt{bx}}$, $x\in(0,\infty)$ and $1\leq b<a$. Then $f$ is increasing on $(0,\infty)$ and therefore by Equation 4,

[TABLE]

with equality if and only if $deg_{G}(v_{1})=2$, $deg_{G}(v_{3})=d_{1}=d_{2}=3$ and $q=d_{3}=d_{4}=4$. 5. 5.

Suppose that $deg_{G}(x_{6})=r$. Then by definition,

[TABLE]

Hence the result. ∎

Lemma 2.2**.**

(See [4])* If $G$ is a connected graph with $n$ vertices and cyclomatic number $\gamma$, then $n_{1}(G)=2-2\gamma+\sum_{i=3}^{\Delta(G)}(i-2)n_{i}$ and $n_{2}(G)=2\gamma+n-2-\sum_{i=3}^{\Delta(G)}(i-1)n_{i}.$*

Corollary 2.3**.**

Let $G$ be a connected graph with $n$ vertices and cyclomatic number $\gamma$.

If $\gamma=5$, then $n_{1}(G)=\sum_{i=3}^{\Delta(G)}(i-2)n_{i}-8$ and $n_{2}(G)=n+8-\sum_{i=3}^{\Delta(G)}(i-1)n_{i}$. 2. 2.

If $\gamma=6$, then $n_{1}(G)=\sum_{i=3}^{\Delta(G)}(i-2)n_{i}-10$ and $n_{2}(G)=n+10-\sum_{i=3}^{\Delta(G)}(i-1)n_{i}$.

Define $\Upsilon_{1}(n)=\{G\mid n_{3}=8,n_{2}=n-8\}$, $\Upsilon_{2}(n)=\{G\mid n_{1}=1,n_{3}=9,n_{2}=n-10\}$, $\Upsilon_{3}(n)$ $=$ $\{G\mid n_{3}=10,n_{2}=n-10\}$ and $\Upsilon_{4}(n)$ $=$ $\{G\mid n_{1}=1,n_{3}=11,n_{2}=n-12\}$.

Corollary 2.4**.**

Let $G$ be an $n-$vertex connected graph, $n\geq 12$, with cyclomatic number $\gamma$ and $\Delta(G)=3$.

If $n_{1}(G)=0$, then $\gamma=5$ if and only if $G\in\Upsilon_{1}(n)$. 2. 2.

If $n_{1}(G)=1$, then $\gamma=5$ if and only if $G\in\Upsilon_{2}(n)$. 3. 3.

If $n_{1}(G)=0$, then $\gamma=6$ if and only if $G\in\Upsilon_{3}(n)$. 4. 4.

If $n_{1}(G)=1$, then $\gamma=6$ if and only if $G\in\Upsilon_{4}(n)$.

Proof.

The proof follows from Corollary 2.3. ∎

Lemma 2.5**.**

Let $G$ be a connected graph with $n$ vertices, $m$ edges and cyclomatic number $\gamma$.

If $n_{1}=0$ and $0<n_{i}<n$ for some $3\leq i\leq n-1$. Then $m_{i,i}(G)\leq n_{i}(T)-2+\gamma$. 2. 2.

If $n_{1}\geq 1$ and $0<n_{i}<n$ for some $3\leq i\leq n-1$. Then $m_{i,i}(G)\leq n_{i}(T)-1+\gamma$.

Proof.

$(1)$ Since $n_{1}=0$, $\gamma(G-v)\leq\gamma(G)-1$, for all $v\in V(G)$. Now the proof follows from this fact that $m\leq n-1+\gamma$. The part $(2)$ is similar. ∎

Let $n$ be a positive integer. Define:

[TABLE]

If $G_{i}\in\Omega_{i}(n)$ for $1\leq i\leq 4$, then $R(G_{1})=\frac{1}{2}n-\frac{5-2\sqrt{6}}{6}$, $R(G_{2})=\frac{1}{2}n-\frac{7-(\sqrt{6}+3\sqrt{2})}{6}$, $R(G_{3})=\frac{1}{2}n-\frac{5-2\sqrt{6}}{6}$ and $R(G_{4})=\frac{1}{2}n-\frac{7-(\sqrt{6}+3\sqrt{2})}{6}$.

Theorem 2.6**.**

The following hold:

Let $G$ be a connected graph with $n\geq 9$ vertices and cyclomatic number 5. Then $R(G)\leq\frac{1}{2}n-\frac{5-2\sqrt{6}}{6}$, with equality if and only if $G\in\Omega_{1}(n)$. 2. 2.

Let $G$ be a connected graph with $n\geq 11$ vertices and cyclomatic number 6. Then $R(G)\leq\frac{1}{2}n-\frac{5-2\sqrt{6}}{6}$, with equality if and only if $G\in\Omega_{3}(n)$.

Proof.

If $G\in\Omega_{1}(n)$, then $R(G)=\frac{1}{2}n-\frac{5-2\sqrt{6}}{6}$, and if $n_{1}(G)=0$, then the result follows from Transformations III, IV and Lemma 2.1(3,4). If $n_{1}(G)\geq 1$, then by Transformations I, II, III, IV, we have $G^{\prime}\in\Omega_{2}(n)$ and by Lemma 2.1(1,2,3,4), $R(G)\leq R(G^{\prime})\leq\frac{1}{2}n-\frac{7-(\sqrt{6}+3\sqrt{2})}{6}<\frac{1}{2}n-\frac{5-2\sqrt{6}}{6}$. 2. 2.

If $G\in\Omega_{3}(n)$, then $R(G)=\frac{1}{2}n-\frac{5-2\sqrt{6}}{6}$, and if $n_{1}(G)=0$, then Transformations III, IV and Lemma 2.1(3,4) gives the result. Finally, if $n_{1}(G)\geq 1$, then by Transformations I, II, III, IV, it follows that $G^{\prime}\in\Omega_{4}(n)$ and by Lemma 2.1(1,2,3,4), $R(G)\leq R(G^{\prime})\leq\frac{1}{2}n-\frac{7-(\sqrt{6}+3\sqrt{2})}{6}<\frac{1}{2}n-\frac{5-2\sqrt{6}}{6}$.

∎

Remark 2.7**.**

Let $G$ be a connected graph with $n=8$ vertices and cyclomatic number 5. Then $R(G)\leq 4$, with equality if and only if $G$ is a $3$-regular graph. 2. 2.

Let $G$ be a connected graph with $n=10$ vertices and cyclomatic number 6. Then $R(G)\leq 5$, with equality if and only if $G$ is a $3$-regular graph.

For positive numbers $k\geq 3$, we define:

[TABLE]

If $G_{1}\in\Lambda_{k}^{1}(n)$ and $G_{2}\in\Gamma_{k}^{1}(n)$, then $R(G_{1})=\frac{1}{2}n-\frac{5-2\sqrt{6}}{6}$ and $R(G_{2})=\frac{1}{2}n-\frac{7-(\sqrt{6}+3\sqrt{2})}{6}$.

A similar proof as Theorem 2.6 proves the following general result:

Theorem 2.8**.**

Let $G$ be a connected graph with $n$ vertices and cyclomatic number $k$, where $k\geq 3$ is a positive ineger.

If $n\geq 2k-1$, then $R(G)\leq\frac{1}{2}n-\frac{5-2\sqrt{6}}{6}$, with equality if and only if $G\in\Lambda_{k}^{1}(n)$. 2. 2.

If $n=2k-2$, then $R(G)\leq\frac{1}{2}n$, with equality if and only if $G$ is a $3$-regular graph.

Let $n$ be a positive number, $\Upsilon_{5}(n):=\{G:n_{1}=0,n_{4}=1,n_{3}=6,n_{2}=n-7\}$ and $\Upsilon_{6}(n):=\{G:n_{1}=0,n_{4}=1,n_{3}=8,n_{2}=n-9\}$. Define :

[TABLE]

If $G_{i}\in\Omega_{5}(n)$ for $1\leq i\leq 4$, then $R(G_{1})=\frac{1}{2}n-\frac{6-(2\sqrt{3}+\sqrt{6})}{3}$,$R(G_{2})=\frac{1}{2}n-\frac{5-2\sqrt{6}}{3}$, $R(G_{3})=\frac{1}{2}n-\frac{6-(2\sqrt{3}+\sqrt{6})}{3}$ and $R(G_{4})=\frac{1}{2}n-\frac{5-2\sqrt{6}}{3}$.

Theorem 2.9**.**

The following hold:

Let $G$ be a connected graph with $n\geq 9$ vertices and cyclomatic number 5. If $G\not\in\Omega_{1}(n)$, then $R(G)\leq\frac{1}{2}n-\frac{6-(2\sqrt{3}+\sqrt{6})}{3}$, with equality if and only if $G\in\Omega_{5}(n)$. 2. 2.

Let $G$ be a connected graph with $n\geq 11$ vertices and cyclomatic number 6. If $G\not\in\Omega_{3}(n)$, then $R(G)\leq\frac{1}{2}n-\frac{6-(2\sqrt{3}+\sqrt{6})}{3}$, with equality if and only if $G\in\Omega_{7}(n)$.

Proof.

If $n_{1}(G)=0$ and $\Delta(G)=3$, then by Transformation III and Lemma 2.1(3), $R(G)\leq\frac{1}{2}n-\frac{5-2\sqrt{6}}{3}<\frac{1}{2}n-\frac{6-(2\sqrt{3}+\sqrt{6})}{3}$. If $n_{1}(G)=0$ and $\Delta(G)\geq 4$, then by Transformations III, IV, V and Lemma 2.1(3,4,5), $R(G)\leq\frac{1}{2}n-\frac{6-(2\sqrt{3}+\sqrt{6})}{3}$, with equality if and only if $G\in\Omega_{5}(n)$. If $n_{1}(G)\geq 1$, then by Transformations I, II, III, IV, and Lemma 2.1(1,2,3,4), $R(G)\leq\frac{1}{2}n-\frac{7-(\sqrt{6}+3\sqrt{2})}{6}<\frac{1}{2}n-\frac{6-(2\sqrt{3}+\sqrt{6})}{3}$. 2. 2.

If $n_{1}(G)=0$ and $\Delta(G)=3$, then by Transformation III and Lemma 2.1(3), $R(G)\leq\frac{1}{2}n-\frac{5-2\sqrt{6}}{3}<\frac{1}{2}n-\frac{6-(2\sqrt{3}+\sqrt{6})}{3}$. If $n_{1}(G)=0$ and $\Delta(G)\geq 4$, then by Transformations III, IV, V and Lemma 2.1(3,4,5), $R(G)\leq\frac{1}{2}n-\frac{6-(2\sqrt{3}+\sqrt{6})}{3}$, with equality if and only if $G\in\Omega_{5}(n)$. If $n_{1}(G)\geq 1$, then by Transformations I, II, III, IV, and Lemma 2.1(1,2,3,4), $R(G)\leq\frac{1}{2}n-\frac{7-(\sqrt{6}+3\sqrt{2})}{6}<\frac{1}{2}n-\frac{6-(2\sqrt{3}+\sqrt{6})}{3}$.

∎

By a simple calculation one can easily see that the Theorem 2.9(1) holds for $n=8$ and Theorem 2.9(2) holds for $n=10$. On the other hand, Theorems 2.6 and 2.9 imply the following result:

Corollary 2.10**.**

The following hold:

Suppose $n\geq 9$. The connected graphs with cyclomatic number 5 in the sets $\Omega_{1}(n)$ and $\Omega_{5}(n)$ have the first and second maximum Randić index among all $n$-vertex connected graphs with cyclomatic number 5, respectively. 2. 2.

Suppose $n\geq 11$. The connected graphs with cyclomatic number 6 in the sets $\Omega_{7}(n)$ and $\Omega_{8}(n)$ have the first and second maximum Randić index among all $n$-vertex connected graphs with cyclomatic number 6, respectively.

Suppose $k\geq 4$ is a positive integer. Define:

[TABLE]

If $H_{1}\in\Lambda_{k}^{2}(n)$ and $H_{2}\in\Gamma_{k}^{2}(n)$, then $R(H_{1})=\frac{1}{2}n-\frac{5-2\sqrt{6}}{3}$ and $R(H_{2})=\frac{1}{2}n-\frac{6-(2\sqrt{3}+\sqrt{6})}{3}$.

Theorem 2.11**.**

Let $G$ be a connected graph with $n$ vertices and cyclomatic number $k\geq 4$.

If $n\geq 2k-1$ and $G\not\in\Lambda_{k}^{1}(n)$, then $R(G)\leq\frac{1}{2}n-\frac{6-(2\sqrt{3}+\sqrt{6})}{3}$, with equality if and only if $G\in\Gamma_{k}^{2}(n)$. 2. 2.

If $n=2k-2$ and $G$ is not a $3$-regular graph, then $R(G)\leq\frac{1}{2}n-\frac{6-(2\sqrt{3}+\sqrt{6})}{3}$, with equality if and only if $G\in\Gamma_{k}^{2}(n)$.

Proof.

The proof is similar to the proof of Theorem 2.6. ∎

We end this paper by the following result that follows from Theorems 2.8 and 2.11.

Theorem 2.12**.**

Let $G$ be a connected graph with $n$ vertices and cyclomatic number $k\geq 4$.

If $n\geq 2k-1$, then the connected graphs with cyclomatic number $k$ in the sets $\Lambda_{k}^{1}(n)$ and $\Gamma_{k}^{2}(n)$ have the first and second maximum Randić index among all $n$-vertices connected graphs with cyclomatic number $k$, respectively. 2. 2.

If $n=2k-2$, then the $3$-regular connected graphs and the connected graphs in the set $\Gamma_{k}^{2}(n)$ with cyclomatic number $k$ have the first and second maximum Randić index among all $n$-vertices connected graphs with cyclomatic number $k$, respectively.

Acknowledgement. The research of the first author is partially supported by the university of Kashan under grant number 890190/5.