On Trees with a Given Number of Vertices of Fixed Degree and Their Two Bond Incident Degree Indices

Abstract

This paper is mainly concerned with the study of two bond incident degree (BID) indices, namely the variable sum exdeg index $SE{I}_a$ and the general zeroth-order Randić index ${}^{0}R_{\alpha }$. The minimum values of $SE{I}_a$ and ${}^{0}R_{\alpha }$ in the class of all trees of fixed order containing no vertex of even degree are obtained for $a>1$ and $\alpha \in [0,1]$; also, the maximum value of ${}^{0}R_{\alpha }$ in the mentioned class is determined for $0<\alpha <1$. Moreover, in the family of all trees of fixed order and with a given number of vertices of even degrees, the extremum values of $SE{I}_a$ and ${}^{0}R_{\alpha }$ are found for every real number $\alpha \notin \{0,1\}$ and $a>1$. Furthermore, in the class of all trees of fixed order and with a given number of vertices of maximum degree, the minimum values of $SE{I}_a$ and ${}^{0}R_{\alpha }$ are determined when $a>1$ and $\alpha$ does not belong to the closed interval $[0,1]$; in the same class, the maximum values of ${}^{0}R_{\alpha }$ are also found for $0<\alpha <1$. The graphs that achieve the obtained extremal values are also determined.

1. Introduction

Let $G=\left(V\right(G),E(G\left)\right)$ be a simple and connected graph with $V\left(G\right)$ and $E\left(G\right)$ denoting its set of vertices and set of edges, respectively. The number of vertices and the number of edges of G are called the order and size of G, respectively. A graph of order n and size m is called an $(n,m)$-graph. The degree of a vertex $x\in V\left(G\right)$, denoted by $d_t$, is the number of vertices adjacent to x. Vertices of degree one are referred to as pendent vertices, while those with a degree greater than two are called branching vertices. For a vertex $x\in V\left(G\right)$, let ${\mathcal{N}}_G\left(x\right)$ denote the collection of all vertices that are neighbors of x, and let ${\mathcal{N}}_G\left[x\right]={\mathcal{N}}_G\left(x\right)\bigcup \left\{x\right\}$. The maximum degree in a graph G is denoted by $\Delta \left(G\right)$. The complement of a graph G is represented by $\overline{G}$ and is defined as the graph with the vertex set $V\left(G\right)$ such that two vertices in $\overline{G}$ are adjacent if and only if they are not adjacent in G. For a graph G, if $A\subseteq E\left(G\right)$ and $B\subseteq E\left(\overline{G}\right)$ then $(G-A)+B$ is the graph obtained from G by removing all the edges belonging to A and inserting all the edges belonging to B. Suppose that we obtain a graph $G^{\ast }$ after applying some graph transformation on G such that $V\left(G^{\ast }\right)=V\left(G\right)$; in this scenario, $d_x$ denote the degree of the vertex x in G. Additional terminology related to the present work can be found in [1].

Graphs can be used to model chemical structures by replacing atoms with vertices and bonds (between atoms) with edges [2]. Such graphs are known as molecular graphs. A topological index of a molecular graph is a numerical value reflecting certain structural features of the corresponding molecule [3] such that it remains the same under graph isomorphism. Topological indices may be helpful in QSPR/QSAR modeling; particularly, in predicting the physicochemical characteristics of chemical species. In this regard, numerous topological indices have been defined to determine different characteristics of molecules; for instance, see [4,5,6].

Many topological indices can be defined via degrees of pairs of vertices incident to edges; such indices are called bond incident degree (BID) indices. Several BID indices have found applications in theoretical chemistry, especially in QSAR and QSPR research [7,8]. Among these useful topological indices, we discuss a few that are relevant to our work.

The first Zagreb index [9] and the Randić index (or the connectivity index) [10] (both emerged in the 1970s within molecular-modeling studies) are among the widely studied indices in the mathematical chemistry literature. The first Zagreb and Randić indices of a graph G are defined as

$$Z_1\left(G\right)=\sum _{u\in V\left(G\right)}{\left(d_u\right)}^2$$

and

$$R\left(G\right)=\sum _{uv\in E\left(G\right)}{\left(d_u{d}_v\right)}^{-1/2},$$

respectively.

The first Zagreb index in general form was put forward in [11,12,13]. This general form of the index $Z_1$ in the literature is known as the general zeroth-order Randić index or the variable first Zagreb index or the first general Zagreb index; in the present paper, we use the former name for the mentioned general index. It is mathematically expressed as ${}^{0}R_{\alpha }\left(G\right)={\sum }_{v\in V\left(G\right)}{\left(d_v\right)}^{\alpha }$, where $\alpha \notin \{0,1\}$ is a real number. From this index, we can recover several additional indices. For instance, the case $\alpha =2$ gives $Z_1$, while $\alpha ={\displaystyle \frac{-1}{2}}$ yields the zeroth-order Randić index [14].

The first three maximum and minimum values of ${}^{0}R_{\alpha }$ for some classes of trees have been studied by Li and Zhao in [15]. The $(n,m)$-graphs with maximum degree at most 4, have been studied for ${}^{0}R_{\alpha }$ in [11]. The index ${}^{0}R_{\alpha }$ for unicyclic graphs has been examined by Zhang and Zhang in [16]. The first three maximum and minimum values of ${}^{0}R_{\alpha }$ are investigated by Zhang et al. [17]. Hu et al. [18], followed by Li and Shi [19], investigated further the index ${}^{0}R_{\alpha }$ for $(n,m)$-graphs. Cheng et al. [20] studied extremal values of the index ${}^{0}R_{\alpha }$ of bipartite graphs by fixing some graph parameters. Su et al. [21] presented several sufficient conditions for graphs to be maximally edge-connected in terms of the index ${}^{0}R_{\alpha }$, and hence generalized the results given by Dankelmann et al. [22]. In [18], extremal values of the topological index ${}^{0}R_{\alpha }$ for connected graphs have been discussed. In [23], sharp lower bounds on ${}^{0}R_{\alpha }$ for connected graphs containing exactly one cycle and with a fixed diameter have been investigated.

The variable sum exdeg index of a graph G was proposed by Vukičević in [24]. This index for a graph G is defined as

$$SE{I}_a\left(G\right)=\sum _{ij\in E\left(G\right)}\left(a^{d_i}+a^{d_j}\right)=\sum _{i\in V\left(G\right)}\left({d_{i}a}^{d_i}\right),$$

where a is a positive real number not equal to 1. This index has a strong correlation with the octane-water partition coefficient of octane isomers [24]. The polynomial form of this topological index was suggested by Yarahmadi and Ashrafi [25]. The index $SE{I}_a$ was further studied in [26,27] for different classes of graphs. Extremal graphs for the index $S{EI}_a$ with certain fixed parameters have been investigated in [28]. Further details about the mathematical studies of this index can be found in the references [29,30].

A graph with n vertices is called an n-vertex graph. The rest of the paper is organized as follows. In the next section, the minimum values of $SE{I}_a$ and ${}^{0}R_{\alpha }$ in the class of all trees of fixed order containing no vertex of even degree are obtained for $a>1$ and $\alpha \in [0,1]$; also, the maximum value of ${}^{0}R_{\alpha }$ in the mentioned class is determined for $0<\alpha <1$. In Section 3, over the family of all trees of fixed order and with a given number of vertices of even degrees, the extremum values of $SE{I}_a$ and ${}^{0}R_{\alpha }$ are found for every real number $\alpha \notin \{0,1\}$ and $a>1$. In Section 4, over the class of all trees of fixed order and with a given number of vertices of maximum degree, the minimum values of $SE{I}_a$ and ${}^{0}R_{\alpha }$ are determined when $a>1$ and $\alpha$ does not belong to the closed interval $[0,1]$; over the same class, the maximum values of ${}^{0}R_{\alpha }$ are also found for $0<\alpha <1$. Section 5 concludes the paper.

2. Extremal Values of ${\mathit{SEI}}_{\mathit{a}}$ and ${}^{\mathbf{0}}\mathit{R}_{\mathit{\alpha }}$ for Trees Containing no Vertex of Even Degree

Denote by $\mathbb{O}{\mathbb{T}}_n$ the class of all those n-vertex trees in which all vertices have odd degree; evidently, n must be even. To establish the main theorem of the current section, we develop a lemma first concerning $SE{I}_a$ and ${}^{0}R_{\alpha }$.

Lemma 1.

Let $\alpha \notin \{0,1\}$ be a real number and $a>1$. In the class $\mathbb{O}{\mathbb{T}}_n$, let T be a tree attaining the following:

(i) 

Minimum $SE{I}_a$ value;

(ii) 

Minimum ${}^{0}R_{\alpha }$ value either for $\alpha <0$ or for $\alpha >1$;

(iii) 

Maximum ${}^{0}R_{\alpha }$ value for $0<\alpha <1$.

Then, the maximum degree of T is three.

Proof. 

Assume to the contrary that the maximum degree of T is different from 3. Then, there is a vertex $x\in V\left(T\right)$ such that $d_x\ge 5$. Suppose that ${\mathcal{N}}_T\left(x\right)=\{b_1,b_2,\cdots ,b_{d_x}\}$. Let w be a pendent vertex of T such that $b_1$ and $b_2$ do not lie on the unique path in T joining x and w. Define $T^{\prime }=(T-\{x{b}_1,x{b}_2\})+\{b_{1}w,b_{2}w\}$ as shown in Figure 1. Clearly, $T^{\prime }\in \mathbb{O}{\mathbb{T}}_n$. Moreover,

$$SE{I}_a\left(T\right)-SE{I}_a\left(T^{\prime }\right)=\left[d_x{a}^{d_x}-(d_x-2)a^{d_x-2}\right]-\left[3a^3-a\right].$$

Since the function $f\left(t\right)=t{a}^t$ is continuous on $[1,d_x]$, the mean value theorem ensures the existence of the real numbers ${\mu }_1\in (1,3)$ and ${\mu }_2\in (d_x-2,d_x)$ such that

$$3a^3-a=2a^{{\mu }_1}(1+{\mu }_{1}lna)$$

and

$$d_x{a}^{d_x}-(d_x-2)a^{d_x-2}=2a^{{\mu }_2}(1+{\mu }_{2}lna).$$

Therefore,

$$SE{I}_a\left(T\right)-SE{I}_a\left(T^{\prime }\right)=2\left[a^{{\mu }_2}\left(1+{\mu }_{2}lna\right)-a^{{\mu }_1}\left(1+{\mu }_{1}lna\right)\right].$$

Note that the function $g\left(t\right)=a^t(1+tlna)$ is increasing on $[1,d_x]$. Since ${\mu }_2>{\mu }_1$, we conclude that $SE{I}_a\left(T\right)-SE{I}_a\left(T^{\prime }\right)>0$. Thus, $SE{I}_a\left(T\right)$ is not minimum in the class $\mathbb{O}{\mathbb{T}}_n$, which contradicts the definition of T.

Figure 1. The trees T and $T^{\prime }=(T-\{x{b}_1,x{b}_2\})+\{b_{1}w,b_{2}w\}$ used in Lemma 1.

We also have

$${}^{0}R_{\alpha }\left(T\right)-{}^{0}R_{\alpha }\left(T^{\prime }\right)=\left[d_x^{\alpha }-{(d_x-2)}^{\alpha }\right]-\left[3^{\alpha }-1\right].$$

There exist real numbers ${\mu }_3\in (1,3)$ and ${\mu }_4\in (d_x-2,d_x)$ such that

$${}^{0}R_{\alpha }\left(T\right)-{}^{0}R_{\alpha }\left(T^{\prime }\right)=2\alpha \left[{\mu }_4^{\alpha -1}-{\mu }_3^{\alpha -1}\right].$$

Thus, if $\alpha >1$ or $\alpha <0$, then ${}^{0}R_{\alpha }\left(T\right)-{}^{0}R_{\alpha }\left(T^{\prime }\right)>0$, and hence ${}^{0}R_{\alpha }\left(T\right)$ is not minimum, which is a contradiction. Similarly, if $0<\alpha <1$, then ${}^{0}R_{\alpha }\left(T\right)-{}^{0}R_{\alpha }\left(T^{\prime }\right)<0$, and so ${}^{0}R_{\alpha }\left(T\right)$ is not maximum, which is a contradiction too. □

Theorem 1.

Let $a>1$ and $\alpha \notin \{0,1\}$ be a real numbers. If $T\in \mathbb{O}{\mathbb{T}}_n$, then

$$SE{I}_a\left(T\right)\ge {\displaystyle \frac{n}{2}}(3a^3+a)-a(3a^2-1),$$

$${}^{0}R_{\alpha }\left(T\right)\ge 1-3^{\alpha }+{\displaystyle \frac{n}{2}}(3^{\alpha }+1)\mathit{when}\mathit{either}\alpha >1\mathrm{or}\alpha <0,$$

and

$${}^{0}R_{\alpha }\left(T\right)\le 1-3^{\alpha }+{\displaystyle \frac{n}{2}}(3^{\alpha }+1)\mathit{when}0<\alpha <1,$$

where any of the equalities holds if and only if T possesses the degree sequence

$$(\underset{{\displaystyle \frac{n}{2}}+1}{\underbrace{1,1,1,\dots ,1,1}},\underset{{\displaystyle \frac{n}{2}}-1}{\underbrace{3,3,\dots ,3,3}}).$$

Proof. 

Let $T^{\ast }\in \mathbb{O}{\mathbb{T}}_n$ be a tree attaining the following

(i)

Minimum $SE{I}_a$ value;

(ii)

Minimum ${}^{0}R_{\alpha }$ value either for $\alpha <0$ or for $\alpha >1$;

(iii)

Maximum ${}^{0}R_{\alpha }$ value for $0<\alpha <1$.

Then, by Lemma 1, the maximum degree of $T^{\ast }$ is three. Let k and $k^{\prime }$ be the numbers of vertices of degrees three and one, respectively, in $T^{\ast }$. Then

$$k+k^{\prime }=n\mathrm{and}3k+k^{\prime }=2(n-1).$$

Hence, we have $k={\displaystyle \frac{n}{2}}-1$ and $n-k={\displaystyle \frac{n}{2}}+1$. Consequently, we obtain

$$SE{I}_a\left(T^{\ast }\right)={\displaystyle \frac{n}{2}}(3a^3+a)-a(3a^2-1)$$

and

$${}^{0}R_{\alpha }\left(T^{\ast }\right)=1-3^{\alpha }+{\displaystyle \frac{n}{2}}(3^{\alpha }+1).$$

□

Example 1.

Consider the graph $T^{\prime }$ given in Figure 2. Clearly $T^{\prime }\in {\mathbb{TO}}_{10}$ and the degree sequence of $T^{\prime }$ is $(1,1,1,1,1,1,3,3,3,3)$. So, by taking $a=2$ and $\alpha =-{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}$, and applying Theorem 1, we obtain

$$SE{I}_2\left(T\right)\ge SE{I}_2\left(T^{\prime }\right)=108,$$

$${}^{0}R_{-{\displaystyle \frac{1}{2}}}\left(T\right)\ge {}^{0}R_{-{\displaystyle \frac{1}{2}}}\left(T^{\prime }\right)={\displaystyle \frac{4\sqrt{3}+18}{3}},$$

and

$${}^{0}R_{{\displaystyle \frac{1}{2}}}\left(T\right)\le {}^{0}R_{{\displaystyle \frac{1}{2}}}\left(T^{\prime }\right)=4\sqrt{3}+6$$

for every 10-vertex tree T containing no vertex of even degree.

Figure 2. The tree $T^{\prime }$ considered in Example 1.

3. Extremal Values of ${\mathit{SEI}}_{\mathit{a}}$ and ${}^{\mathbf{0}}\mathit{R}_{\mathit{\alpha }}$ for Trees with $\mathit{r}$ Even Degree Vertices

For integers n and r with $1\le r\le n-4$, we denote by $\mathbb{E}{\mathbb{T}}_{n,r}$ the family of n-vertex trees in which exactly r vertices have even degrees. This section provides the minimum and maximum values of $SE{I}_a$ and ${}^{0}R_{\alpha }$ for the graphs belonging to the class $\mathbb{E}{\mathbb{T}}_{n,r}$, when $\alpha \notin \{0,1\}$ and $a>1$.

Lemma 2.

Let $\alpha \notin \{0,1\}$ be a real number and $a>1$. In the class $\mathbb{E}{\mathbb{T}}_{n,r}$, let T be a tree attaining the following:

(i) 

Minimum $SE{I}_a$ value;

(ii) 

Minimum ${}^{0}R_{\alpha }$ value either for $\alpha <0$ or for $\alpha >1$;

(iii) 

Maximum ${}^{0}R_{\alpha }$ value for $0<\alpha <1$.

 Then, the maximum degree in T is three.

Proof. 

Assume to the contrary that the maximum degree of T is different from three. Then, there is a vertex $x\in V\left(T\right)$ such that $d_x\ge 4$. Let $b_1,b_2\in {\mathcal{N}}_T\left(x\right)$. Let w be a pendent vertex of T such that $b_1$ and $b_2$ do not lie on the unique path in T joining x and w. First, we assume $d_x=4$. Define $T^{\prime }=(T-\left\{x{b}_1\right\})+\left\{b_{1}w\right\}$, as shown in Figure 3. Clearly, $T^{\prime }\in \mathbb{E}{\mathbb{T}}_{n,r}$. Moreover, we have

$$\begin{array}{cc}\hfill SE{I}_a\left(T\right)-SE{I}_a\left(T^{\prime }\right)& =4a^4-3a^3-2a^2+a\hfill \\ \hfill & =a(a-1)(4a^2+a-1)>0.\hfill \end{array}$$

So, T does not minimize $SE{I}_a$; this contradicts the given assumption on the minimality of $SE{I}_a\left(T\right)$. Similarly,

$${}^{0}R_{\alpha }\left(T\right)-{}^{0}R_{\alpha }\left(T^{\prime }\right)=4^{\alpha }-3^{\alpha }-2^{\alpha }+1.$$

There are real numbers ${\mu }_5\in (1,2)$ and ${\mu }_6\in (3,4)$ such that

$${}^{0}R_{\alpha }\left(T\right)-{}^{0}R_{\alpha }\left(T^{\prime }\right)=\alpha \left[{\mu }_6^{\alpha -1}-{\mu }_5^{\alpha -1}\right].$$

Therefore, when $\alpha >1$ or $\alpha <0$, we obtain that ${}^{0}R_{\alpha }\left(T\right)-{}^{0}R_{\alpha }\left(T^{\prime }\right)>0$, and thus, T does not minimize ${}^{0}R_{\alpha }$. Similarly, if $0<\alpha <1$, then ${}^{0}R_{\alpha }\left(T\right)-{}^{0}R_{\alpha }\left(T^{\prime }\right)<0$, and so T does not maximize ${}^{0}R_{\alpha }$. In any case concerning $\alpha$, we arrive at a contradiction due to the given assumption on ${}^{0}R_{\alpha }\left(T\right)$.

Now, we consider the case when $d_x\ge 5$. Define $T^{″}=(T-\{x{b}_1,x{b}_2\})+\{b_{1}w,b_{2}w\}$. Clearly, $T^{\prime \prime }\in \mathbb{E}{\mathbb{T}}_{n,r}$. The rest of the proof is very similar to the proof of Lemma 1, and hence, we omit it. □

Figure 3. The trees T and $T^{\prime }=(T-\left\{x{b}_1\right\})+\left\{b_{1}w\right\}$ considered in Lemma 2.

Theorem 2.

Let $a>1$ and $\alpha \notin \{0,1\}$ be real numbers. If $T\in \mathbb{E}{\mathbb{T}}_{n,r}$, then

$$SE{I}_a\left(T\right)\ge {\displaystyle \frac{a(3a^2+1)n+a(4a-1-3a^2)r+2a(1-3a^2)}{2}},$$

$${}^{0}R_{\alpha }\left(T\right)\ge 3^{\alpha }\left[{\displaystyle \frac{n-2-r}{2}}\right]+2^{\alpha }r+{\displaystyle \frac{n+2-r}{2}}\mathit{when}\mathit{either}\alpha <0\mathrm{or}\alpha >1,$$

and

$${}^{0}R_{\alpha }\left(T\right)\le 3^{\alpha }\left[{\displaystyle \frac{n-2-r}{2}}\right]+2^{\alpha }r+{\displaystyle \frac{n+2-r}{2}}\mathit{when}0<\alpha <1.$$

Moreover, the equality in any of the three inequalities holds if and only if T has the degree sequence

$$(\underset{{\displaystyle \frac{n-r+2}{2}}}{\underbrace{1,1,1\dots ,1,1}}\underset{r}{\underbrace{2,2,\dots ,2,2,}}\underset{{\displaystyle \frac{n-r-2}{2}}}{\underbrace{3,3,\dots ,3,3}}).$$

Proof. 

Let $T^{\ast }\in \mathbb{E}{\mathbb{T}}_{n,r}$ be a tree attaining the following:

(i)

Minimizing $SE{I}_a$

(ii)

Minimizing ${}^{0}R_{\alpha }$ either for $\alpha <0$ or for $\alpha >1$

(iii)

Maximizing ${}^{0}R_{\alpha }$ for $0<\alpha <1$.

Then, by Lemma 2, the maximum degree of $T^{\ast }$ is three. For $i=1,2,3$, let $n_i$ be the number of vertices of degree i in $T^{\ast }$. Then,

$$n_2=r,n_1+n_3=n-r,\mathrm{and}{n}_1+3n_3=2n-2r-2.$$

Hence, we obtain

$$n_3={\displaystyle \frac{n-r-2}{2}}\mathrm{and}{n}_1={\displaystyle \frac{n-r+2}{2}}.$$

Therefore,

$$SE{I}_a\left(T^{\ast }\right)={\displaystyle \frac{a(3a^2+1)n+a(4a-1-3a^2)r+2a(1-3a^2)}{2}}$$

and

$${}^{0}R_{\alpha }\left(T^{\ast }\right)=3^{\alpha }\left[{\displaystyle \frac{n-2-r}{2}}\right]+2^{\alpha }r+{\displaystyle \frac{n+2-r}{2}}.$$

□

Lemma 3.

Let $\alpha \notin \{0,1\}$ be a real number and $a>1$. In the class $\mathbb{E}{\mathbb{T}}_{n,r}$, let T be a tree attaining the following:

(i) 

Maximum $SE{I}_a$ value;

(ii) 

Maximum ${}^{0}R_{\alpha }$ value either for $\alpha <0$ or for $\alpha >1$;

(iii) 

Minimum ${}^{0}R_{\alpha }$ value for $0<\alpha <1$.

 Then, T has at most one vertex of degree more than 2.

Proof. 

For the sake of contradiction, we assume that T has at least two vertices of degree larger than 2. Let x and y be two vertices of T such that $d_x\ge d_y\ge 3$. Let $w_1,\dots ,w_{d_y-2},w_{d_y-1}$ be the neighboring vertices of y that do not lie on the unique path joining x and y. We deal with two cases.

Case 1. Either $d_x$ or $d_y$ is even.

For this case, define $T^{\prime }=(T-\{w_{1}y,\dots ,w_{d_y-2}y\})+\{w_{1}x,\dots ,w_{d_y-2}x\}$, as shown in Figure 4. We can see that $T^{\prime }\in \mathbb{E}{\mathbb{T}}_{n,r}$. Moreover,

$$SE{I}_a\left(T\right)-SE{I}_a\left(T^{\prime }\right)=\left[d_x{a}^{a_x}-(d_x+d_y-2)a^{d_x+d_y-2}\right]+\left[y{a}^y-2a^2\right]$$

By the mean value theorem, there are numbers ${\mu }_1\in (2,d_y)$ and ${\mu }_2\in (d_x,d_x+d_y-2)$ such that

$$y{a}^y-2a^2=(d_y-2)a^{{\mu }_1}(1+{\mu }_{1}lna)$$

and

$$(d_x+d_y-2)a^{d_x+d_y-2}-x^{d}x=(d_y-2)a^{{\mu }_2}(1+{\mu }_{2}lna).$$

Thus,

$$SE{I}_a\left(T\right)-SE{I}_a\left(T^{\prime }\right)=(d_y-2)\left[a^{{\mu }_1}(1+{\mu }_{1}lna)-a^{{\mu }_2}(1+{\mu }_{2}lna)\right]<0.$$

Hence, $SE{I}_a\left(T\right)$ is not the maximum in the considered class of trees, which yields a contradiction. Similarly, there are numbers ${\mu }_3\in (2,d_y)$ and ${\mu }_4\in (d_x,d_x+d_y-2)$ such that

$$\begin{array}{cc}\hfill {}^{0}R_{\alpha }\left(T\right)-{}^{0}R_{\alpha }\left(T^{\prime }\right)& =\left[d_x^{\alpha }-{(d_x+d_y-2)}^{\alpha }\right]+\left[d_y^{\alpha }-2^{\alpha }\right].\hfill \\ \hfill & =(d_y-2)\alpha \left[{\mu }_3^{\alpha -1}-{\mu }_4^{\alpha -1}\right].\hfill \end{array}$$

Therefore, we obtain

$${}^{0}R_{\alpha }\left(T\right)>{}^{0}R_{\alpha }\left(T^{\prime }\right)\mathrm{when}\mathrm{either}\alpha <0\mathrm{or}\alpha >1,$$

and

$${}^{0}R_{\alpha }\left(T\right)<{}^{0}R_{\alpha }\left(T^{\prime }\right)\mathrm{when}0<\alpha <1,$$

which again yields a contradiction.

Figure 4. The trees T and $T^{\prime }=(T-\{w_{1}y,\dots ,w_{d_y-2}y\})+\{w_{1}x,\dots ,w_{d_y-2}x\}$ considered in Case 1 of Lemma 3.

Case 2. Both $d_x$ and $d_y$ are odd.

For this case, define $T^{″}=(T-\{w_{1}y,\dots ,w_{d_y-1}y\})+\{w_{1}x,\dots ,w_{d_y-1}x\}$, as shown in Figure 5. We can see that $T^{″}\in \mathbb{E}{\mathbb{T}}_{n,r}$. Moreover,

$$SE{I}_a\left(T\right)-SE{I}_a\left(T^{″}\right)=\left[d_x{a}^{a_x}-(d_x+d_y-1)a^{d_x+d_y-1}\right]+\left[d_y{a}^{d_y}-a\right]$$

and

$${}^{0}R_{\alpha }\left(T\right)-{}^{0}R_{\alpha }\left(T^{″}\right)=\left[d_x^{\alpha }-{(d_x+d_y-1)}^{\alpha }\right]+\left[d_y^{\alpha }-1\right].$$

By the mean value theorem, there are numbers ${\nu }_1,{\nu }_3\in (1,d_y)$ and ${\nu }_2,{\nu }_4\in (d_x,d_x+d_y-1)$ such that

$$SE{I}_a\left(T\right)-SE{I}_a\left(T^{″}\right)=(d_y-1)\left[a^{{\nu }_1}(1+{\nu }_{1}lna)-a^{{\nu }_2}(1+{\nu }_{2}lna)\right]<0.$$

and

$${}^{0}R_{\alpha }\left(T\right)-{}^{0}R_{\alpha }\left(T^{″}\right)=(d_y-1)\alpha \left[{\nu }_3^{\alpha -1}-{\nu }_4^{\alpha -1}\right].$$

Thus, we have

$$SE{I}_a\left(T\right)>SE{I}_a\left(T^{″}\right),$$

$${}^{0}R_{\alpha }\left(T\right)>{}^{0}R_{\alpha }\left(T^{″}\right)\mathrm{when}\mathrm{either}\alpha <0\mathrm{or}\alpha >1,$$

and

$${}^{0}R_{\alpha }\left(T\right)<{}^{0}R_{\alpha }\left(T^{″}\right)\mathrm{when}0<\alpha <1,$$

which yields a contradiction. Hence, T has at most one vertex of degree larger than two. □

Figure 5. The trees T and $T^{″}=(T-\{w_{1}y,\dots ,w_{d_y-1}y\})+\{w_{1}x,\dots ,w_{d_y-1}x\}$ considered in Case 2 of Lemma 3.

Theorem 3.

Let $a>1$ and $\alpha \notin \{0,1\}$ be real numbers. If $T\in \mathbb{E}{\mathbb{T}}_{n,r}$ then

$$SE{I}_a\left(T\right)\le (n-r)a^{n-r}+(2a^2-a)r+(n-2a)a,$$

$${}^{0}R_{\alpha }\left(T\right)\le {(n-r)}^{\alpha }+(2^{\alpha }-1)r+n-2^{\alpha }\mathit{when}\mathit{either}\alpha <0\mathrm{or}\alpha >1,$$

and 

$${}^{0}R_{\alpha }\left(T\right)\ge {(n-r)}^{\alpha }+(2^{\alpha }-1)r+n-2^{\alpha }\mathit{when}0<\alpha <1.$$

Moreover, the equality in any of the three inequalities holds if and only if T has the following degr- ee sequence:

$$(\underset{n-r}{\underbrace{1,1,1\dots ,1,1}},\underset{r-1}{\underbrace{2,2,\dots ,2,2}},n-r).$$

Proof. 

Among all trees in the class $\mathbb{E}{\mathbb{T}}_{n,r}$, let $T^{\prime }$ be a tree attaining the following:

(i)

Maximizing $SE{I}_a$;

(ii)

Maximizing ${}^{0}R_{\alpha }$ either for $\alpha <0$ or for $\alpha >1$;

(iii)

Minimizing ${}^{0}R_{\alpha }$ for $0<\alpha <1$.

Then, by Lemma 3, $T^{\prime }$ has at most one vertex of degree larger than two. Choose $x\in V\left(T^{\prime }\right)$ such that $d_x\ge 3$. For $i\in 1,2$, let $n_i$ denote the number of vertices of degree i in $T^{\prime }$. Then, we obtain $d_x=n_1=n-n_2-1$. So, we obtain

$$SE{I}_a\left(T^{\prime }\right)=(n-n_2-1)a+2n_2{a}^2+(n-n_2-1)a^{n-n_2-1}.$$

If $d_x$ is odd, then $n_2=r$, yielding

$$SE{I}_a\left(T^{\prime }\right)=(n-r-1)a+2r{a}^2+(n-r-1)a^{n-r-1}.$$

However, if $d_x$ is even, then $n_2=r-1$, which implies

$$SE{I}_a\left(T^{\prime }\right)=(n-r)a+2(r-1)a^2+(n-r)a^{n-r}.$$

Since

$$(n-r-1)a+2r{a}^2+(n-r-1)a^{n-r-1}<(n-r)a+2(r-1)a^2+(n-r)a^{n-r},$$

we conclude that $d_x$ is even. Hence, $n_1=n-r=d_x$ and $n_2=r-1$. Therefore, we have

$$SE{I}_a\left(T^{\prime }\right)=(n-r-1)a+2r{a}^2+(n-r-1)a^{n-r-1}.$$

Similarly, we have

$${}^{0}R_{\alpha }\left(T^{\prime }\right)=(n-n_2-1)+n_2{2}^{\alpha }+{(n-n_2-1)}^{\alpha }.$$

Now, if either $\alpha <0$ or $\alpha >1$, then

$$(n-r-1)+r{2}^{\alpha }+{(n-r-1)}^{\alpha }<(n-r)+(r-1)2^{\alpha }+{(n-r)}^{\alpha },$$

and when $0<\alpha <1$, then

$$(n-r-1)+r{2}^{\alpha }+{(n-r-1)}^{\alpha }>(n-r)+(r-1)2^{\alpha }+{(n-r)}^{\alpha }.$$

Again, we establish that $n_1=n-r=d_x$ and $n_2=r-1$, and hence

$${}^{0}R_{\alpha }\left(T^{\prime }\right)=(n-r)+(r-1)2^{\alpha }+{(n-r)}^{\alpha }.$$

□

4. Extremal Values of ${\mathit{SEI}}_{\mathit{a}}$ and ${}^{\mathbf{0}}\mathit{R}_{\mathit{\alpha }}$ for Trees with  $\mathit{k}$  Vertices of Maximum Degree

For integers k and n such that $1\le k\le n-2$, denote by $\mathbb{M}{\mathbb{T}}_{n,k}$ the class of all n-vertex tree graphs that have exactly k vertices of maximum degree. In this section, we determine the minimum values of $SE{I}_a$ and ${}^{0}R_{\alpha }$ over the class $\mathbb{M}{\mathbb{T}}_{n,k}$ when $a>1$ and $\alpha$ does not belong to the closed interval $[0,1]$. We also determine the maximum value of ${}^{0}R_{\alpha }$ over the class $\mathbb{M}{\mathbb{T}}_{n,k}$ when $0<\alpha <1$. Note that the class $\mathbb{M}{\mathbb{T}}_{n,n-2}$ consists of only one tree, namely the path graph $P_n$. Also, we remark that the class $\mathbb{M}{\mathbb{T}}_{n,k}$ is empty whenever ${\displaystyle \frac{n}{2}}-1<k<n-2$. Hence, in the rest of this section, we assume that $1\le k\le {\displaystyle \frac{n}{2}}-1$.

Lemma 4.

Let $\alpha \notin \{0,1\}$ be a real number and $a>1$. In the class $\mathbb{M}{\mathbb{T}}_{n,k}$, let T be a tree attaining the following:

(i) 

Minimum $SE{I}_a$ value;

(ii) 

Minimum ${}^{0}R_{\alpha }$ value either for $\alpha <0$ or for $\alpha >1$;

(iii) 

Maximum ${}^{0}R_{\alpha }$ value for $0<\alpha <1$.

 Then, the maximum degree of a vertex in T is three.

Proof. 

Assume that the maximum degree of T is $\Delta$. For the sake of contradiction, we suppose that $\Delta \ge 4$. Choose $x\in V\left(T\right)$ such that $d_x=\Delta$. Let w be a pendent vertex of T. Let y be a neighbor of x not lying on the unique path joining x and w. We define

$$T^{\left(1\right)}=(T-\left\{xy\right\})+\left\{yw\right\}$$

(1)

as shown in Figure 6. We say that $T^{\left(1\right)}$ is obtained from T by applying the transformation 1 on the vertex x of maximum degree $\Delta$. Then, we have

$$SE{I}_a\left(T^{\left(1\right)}\right)-SE{I}_a\left(T\right)=\left[2a^2-a\right]-\left[\left(\Delta \right)a^{\Delta }-(\Delta -1)a^{\Delta -1}\right]$$

and

$${}^{0}R_{\alpha }\left(T^{\left(1\right)}\right)-{}^{0}R_{\alpha }\left(T\right)=\left[{\left(2\right)}^{\alpha }-{\left(1\right)}^{\alpha }\right]-\left[{\left(\Delta \right)}^{\alpha }-{\left(\Delta -1\right)}^{\alpha }\right].$$

There exist the numbers ${\mu }_1,{\mu }_3\in (1,2)$ and ${\mu }_2,{\mu }_4\in (\Delta -1,\Delta )$ such that

$$SE{I}_a\left(T^{\left(1\right)}\right)-SE{I}_a\left(T\right)=\left[a^{{\mu }_1}\left(1+{\mu }_{1}lna\right)-a^{{\mu }_2}\left(1+{\mu }_{2}lna\right)\right]$$

and

$${}^{0}R_{\alpha }\left(T^{\left(1\right)}\right)-{}^{0}R_{\alpha }\left(T\right)=\alpha \left[{\mu }_3^{\alpha -1}-{\mu }_4^{\alpha -1}\right].$$

From the inequalities ${\mu }_1<{\mu }_2$ and ${\mu }_3<{\mu }_4$, it follows that

$$SE{I}_a\left(T\right)>SE{I}_a\left(T^{\left(1\right)}\right),$$

$${}^{0}R_{\alpha }\left(T^{\left(1\right)}\right)<{}^{0}R_{\alpha }\left(T\right)\mathrm{when}\mathrm{either}\alpha <0\mathrm{or}\alpha >1,$$

and

$${}^{0}R_{\alpha }\left(T^{\left(1\right)}\right)>{}^{0}R_{\alpha }\left(T\right)\mathrm{when}0<\alpha <1.$$

Now, the tree $T^{\left(1\right)}$ has $k-1$ vertices of degree $\Delta$. Let t be the number of vertices of degree $\Delta -1$ in T. After applying the transformation 1 successively $k+t$ times, k times on vertices of degree $\Delta$ and t times on vertices of degree $\Delta -1$, we construct a tree $T^{(k+t)}$ of maximum degree $\Delta -1$. Certainly, $T^{(k+t)}$ has exactly k vertices of degree $\Delta -1$, and hence, $T^{(k+t)}\in \mathbb{M}{\mathbb{T}}_{n,k}$. On the other hand, we have

$$SE{I}_a\left(T^{(k+t)}\right)<SE{I}_a\left(T\right),$$

$${}^{0}R_{\alpha }\left(T^{(k+t)}\right)<{}^{0}R_{\alpha }\left(T\right)\mathrm{when}\mathrm{either}\alpha >0\mathrm{or}\alpha >1,$$

$${}^{0}R_{\alpha }\left(T^{(k+t)}\right)<{}^{0}R_{\alpha }\left(T\right)\mathrm{when}0<\alpha <1,$$

which is a contradiction. Therefore, the maximum degree of T is three. □

Figure 6. The trees T and $T^{\left(1\right)}=(T-\left\{xy\right\})+\left\{yw\right\}$ considered in the proof of Lemma 4.

Theorem 4.

Let $a>1$ and $\alpha \notin \{0,1\}$ be real numbers. If $T\in \mathbb{M}{\mathbb{T}}_{n,k}$ with $1\le k\le {\displaystyle \frac{n}{2}}-1$ then

$$SE{I}_a\left(T\right)\ge 3k{a}^3+2(n-2k-2)a^2+a(k+2),$$

$${}^{0}R_{\alpha }\left(T\right)\ge k{3}^{\alpha }+(n-2k-2)2^{\alpha }+k+2\mathit{when}\mathit{either}\alpha <0\mathrm{or}\alpha >1,$$

and

$${}^{0}R_{\alpha }\left(T\right)\le k{3}^{\alpha }+(n-2k-2)2^{\alpha }+k+2\mathit{when}0<\alpha <1.$$

The equality in any of the above three inequalities holds if and only if T has the following degree sequence:

$$(\underset{k}{\underbrace{3,3,\dots ,3,3,}}\underset{n-2k-2}{\underbrace{2,2,\dots ,2,2,}}\underset{k+2}{\underbrace{1,1,1\dots ,1,1}}).$$

(2)

Proof. 

Let $T^{\prime }\in \mathbb{M}{\mathbb{T}}_{n,k}$ be a tree attaining the following:

(i)

Minimum $SE{I}_a$ value;

(ii)

Minimum ${}^{0}R_{\alpha }$ value either for $\alpha <0$ or for $\alpha >1$;

(iii)

Maximum ${}^{0}R_{\alpha }$ value for $0<\alpha <1$.

Then, by Lemma 4, the maximum degree of $T^{\prime }$ is three. For $i\in \{1,2,3\}$, let $n_i$ be the number of vertices in $T^{\prime }$ of degree i. Since $n_3=k$, it results that $n_1=k+2$, and so $n_2=n-2k-2$. Therefore, $T^{\prime }$ has the degree sequence given in (2). Moreover,

$$SE{I}_a\left(T^{\prime }\right)=3k{a}^3+2(n-2k-2)a^2+a(k+2)$$

and

$${}^{0}R_{\alpha }\left(T^{\prime }\right)=k{3}^{\alpha }+(n-2k-2)2^{\alpha }+k+2.$$

□

Corollary 1.

Let n, k, and l be integers such that $1\le l<k\le {\displaystyle \frac{n}{2}}-1$. If $T^k$ and $T^l$ are the trees attaining the minimum values of $SE{I}_a$ for $a>1$, maximum values of ${}^{0}R_{\alpha }$ for $0<\alpha <1$, and the minimum values of ${}^{0}R_{\alpha }$ for $\alpha \notin [0,1]$, in the classes $\mathbb{M}{\mathbb{T}}_{n,k}$ and $\mathbb{M}{\mathbb{T}}_{n,l}$, respectively, then

$$SE{I}_a\left(T^k\right)>SE{I}_a\left(T^l\right)\mathrm{for}a>1,$$

$${}^{0}R_{\alpha }\left(T^{\left(k\right)}\right)>{}^{0}R_{\alpha }\left(T^{\left(l\right)}\right)\mathit{when}\mathit{either}\alpha <0\mathrm{or}\alpha >1,$$

and

$${}^{0}R_{\alpha }\left(T^{\left(k\right)}\right)<{}^{0}R_{\alpha }\left(T^{\left(l\right)}\right)\mathit{when}0<\alpha <1.$$

5. Conclusions

In this paper, we have addressed the problem of establishing the best possible bounds for two BID indices, namely the variable sum exdeg index $SE{I}_a$ and the general zeroth-order Randić index ${}^{0}R_{\alpha }$, over certain classes of trees. First, we have considered the class of all trees of fixed order containing no vertex of even degree and obtained minimum values of $SE{I}_a$ and ${}^{0}R_{\alpha }$ in the mentioned class for $a>1$ and $\alpha \in [0,1]$; we have also determined the maximum value of ${}^{0}R_{\alpha }$ in this class for $0<\alpha <1$ (see Theorem 1). Additionally, in the family of all trees of fixed order and with a given number of vertices of even degrees, we have found the minimum and maximum values of $SE{I}_a$ and ${}^{0}R_{\alpha }$ for every real number $\alpha \notin \{0,1\}$ and $a>1$ (see Theorems 2 and 3). Furthermore, in the class of all trees of fixed order and with a given number of vertices of maximum degree, we have determined the minimum values of $SE{I}_a$ and ${}^{0}R_{\alpha }$ when $a>1$ and $\alpha$ does not belong to the closed interval $[0,1]$; in the same class, we have also found the maximum value of ${}^{0}R_{\alpha }$ for $0<\alpha <1$ (see Theorem 4). One of the possible research directions toward which the present study may be extended is to extend/generalize the bounds obtained in this paper for other BID indices.