Sharp Upper and Lower Bounds of VDB Topological Indices of Digraphs

Abstract

A vertex-degree-based (VDB, for short) topological index $\phi$ induced by the numbers $\left\{{\phi }_{ij}\right\}$ was recently defined for a digraph D, as ${\displaystyle \phi \left(D\right)=\frac{1}{2}\sum _{uv}{\phi }_{d_u^{+}{d}_v^{-}},}$ where $d_u^{+}$ denotes the out-degree of the vertex $u,$ $d_v^{-}$ denotes the in-degree of the vertex $v,$ and the sum runs over the set of arcs $uv$ of D. This definition generalizes the concept of a VDB topological index of a graph. In a general setting, we find sharp lower and upper bounds of a symmetric VDB topological index over ${\mathcal{D}}_n$, the set of all digraphs with n non-isolated vertices. Applications to well-known topological indices are deduced. We also determine extremal values of symmetric VDB topological indices over $\mathcal{OT}\left(n\right)$ and $\mathcal{O}\left(G\right)$, the set of oriented trees with n vertices, and the set of all orientations of a fixed graph G, respectively.

1. Introduction

A digraph D is a finite nonempty set V called vertices, together with a set A of ordered pairs of distinct vertices of D, called arcs. If $a=(u,v)$ is an arc of D, then we write $uv$ and say that the two vertices are adjacent. Given a vertex u of G, the out-degree of u is denoted by $d_u^{+}$ and defined as the number of arcs of the form $uv$, where $v\in V$. The in-degree of u is denoted by $d_u^{-}$ and defined as the number of arcs of the form $wu$, where $w\in V$. A vertex u in D is called a sink vertex (resp. source vertex) if $d_u^{+}=0$ (resp. $d_u^{-}=0$). We denote by $q=q\left(D\right)$ the number of vertices of D which are sink vertices or source vertices. If $d_u^{+}=d_u^{-}=0$, then u is an isolated vertex. The set of digraphs with n non-isolated vertices is denoted by ${\mathcal{D}}_n$.

One special class of digraphs is the oriented graphs. A pair of arcs of a digraph D of the form $uv$ and $vu$ are called symmetric arcs. If D has no symmetric arcs, then D is an oriented graph. We note that D can be obtained from a graph G by substituting each edge $uv$ by an arc $uv$ or $vu$, but not both. In this case, we say that D is an orientation of G. For example, in Figure 1 we show the directed path ${\overrightarrow{P}}_n$ and the directed cycle ${\overrightarrow{C}}_n$, orientations of the path $P_n$ and cycle $C_n$, respectively. A sink-source orientation of a graph G is an orientation in which every vertex is a sink vertex or a source vertex. Clearly, when we reverse the orientations of all arcs in a sink-source orientation, we obtain a sink-source orientation again. For instance, the digraphs ${\overrightarrow{K}}_{1,n-1}$ and ${\overrightarrow{K}}_{n-1,1}$ in Figure 1 are sink-source orientations of the star $S_n$. Note that ${\overrightarrow{K}}_{n-1,1}$ is obtained by reversing all arcs of ${\overrightarrow{K}}_{1,n-1}$.

Figure 1. Orientations of $P_n$, $C_n$, and $S_n$.

Let $D_1=(V_1,A_1)$ and $D_2=(V_2,A_2)$ be digraphs with no common vertices. The direct sum of digraphs $D_1$ and $D_2$, denoted by $D_1\oplus D_2$, is the digraph with vertex and arc sets $V_1\cup V_2$ and $A_1\cup A_2$, respectively. In general, ${\oplus }_{i=1}^k{D}_i$ denote the direct sum of the digraphs $D_1=(V_1,A_1),\cdots ,D_k=(V_k,A_k)$. If $D_i=D$ for all i, then we simply write ${\oplus }_{i=1}^k{D}_i=kD$.

The following notation and concepts were introduced in [1]. Let $D\in {\mathcal{D}}_n$. Let us denote by $n_i^{+}$ (resp. $n_i^{-}$) the number of vertices in D with out-degree (resp. in-degree) i, for all $0\le i\le n-1$. For every $1\le i,j\le n-1$, define the set

$$A_{ij}=\left\{uv\in A:d_u^{+}=i\mathrm{and}{d}_v^{-}=j\right\}.$$

The cardinality of $A_{ij}$ is denoted by $a_{ij}$. Clearly,

$$\begin{array}{cccccc}\hfill {\displaystyle \sum _{1\le i,j\le n-1}}{a}_{ij}=a& ;& {\displaystyle \sum _{j=1}^{n-1}}{a}_{ij}=i{n}_i^{+}\hfill & ;& and& {\displaystyle \sum _{i=1}^{n-1}}{a}_{ij}=j{n}_j^{-},\end{array}$$

(1)

where a is the number of arcs D has.

A VDB topological index is a function $\phi$ induced by real numbers $\left\{{\phi }_{ij}\right\}$, where $1\le i,j\le n-1$, defined as [1]

$$\phi \left(D\right)=\frac{1}{2}\sum _{1\le i,j\le n-1}{a}_{ij}{\phi }_{ij}.$$

(2)

Equivalently,

$$\phi \left(D\right)=\frac{1}{2}\sum _{uv\in A}{\phi }_{d_u^{+}{d}_v^{-}}.$$

(3)

When ${\phi }_{ij}={\phi }_{ji}$ for all $1\le i,j\le n-1$, we say that $\phi$ is a symmetric VDB topological index. In this case, the expression given in (2) can be simplified. In fact, let

$$p_{ij}=a_{ij}+a_{ji},$$

(4)

for all $1\le i,j\le n-1$, and

$$p_{ii}=a_{ii},$$

(5)

for all $i=1,\dots ,n-1$. Then

$$\phi \left(D\right)=\frac{1}{2}\sum _{\left(i,j\right)\in K}{p}_{ij}{\phi }_{ij},$$

(6)

where

$$K=\left\{\left(i,j\right)\in \mathbb{N}\times \mathbb{N}:1\le i\le j\le n-1\right\}.$$

In particular, when $D=G$ is a graph, it was shown in [1] that Formula (6) reduces to

$$\phi \left(G\right)=\sum _{\left(i,j\right)\in K}{m}_{ij}{\phi }_{ij},$$

where $m_{ij}$ is the number of edges in G which join vertices of degree i and j. So we recover the degree-based-topological indices of graphs, a concept which has been, and currently is, extensively investigated in the mathematical and chemical literature [2,3,4]. For recent results, we refer to [5,6,7,8,9,10,11,12].

This paper is organized as follows. In Section 2, in a general setting (Theorem 1), we find sharp lower and upper bounds of a symmetric VDB topological index over the set ${\mathcal{D}}_n$. As a byproduct, we obtain over ${\mathcal{D}}_n,$ sharp upper and lower bounds of well-known VDB topological indices, which include the First Zagreb index ${\mathcal{M}}_1$ (${\phi }_{ij}=i+j$) [13], the Second Zagreb index ${\mathcal{M}}_2$ (${\phi }_{ij}=ij$) [13], the Randić index $\chi$ (${\phi }_{ij}=1/\sqrt{ij}$) [14], the Harmonic index $\mathcal{H}$ (${\phi }_{ij}=2/(i+j)$) [15], the Geometric-Arithmetic $\mathcal{GA}$ (${\phi }_{ij}=2\sqrt{ij}/(i+j)$) [16], the Sum-Connectivity $\mathcal{SC}$ (${\phi }_{ij}=1/\sqrt{i+j}$) [17], the Atom-Bond-Connectivity $\mathcal{ABC}$ (${\phi }_{ij}=\sqrt{(i+j-2)/ij}$) [18], and the Augmented Zagreb $\mathcal{AZ}$ (${\phi }_{ij}={(ij/(i+j-2))}^3$) [19].

In Section 3, based on Theorem 2, we give sharp upper and lower bounds of symmetric VDB topological indices over the set $\mathcal{OT}\left(n\right)$, the set of oriented trees with n vertices. In particular, we deduce sharp upper and lower bounds for the well-known indices mentioned above over $\mathcal{OT}\left(n\right)$. Finally, in Section 4, we consider the problem of finding the extremal values of a symmetric VDB topological index among all orientations in $\mathcal{O}\left(G\right)$, the set of all orientations of a fixed graph G. In order to do this, we define strictly nondecreasing (resp. nonincreasing) symmetric VDB topological indices and show that for these indices, the value of any orientation at G is not greater (resp. smaller) than half the value at G. Moreover, equality occurs, and only if the orientation is a sink-source orientation of G. In particular, when G is a bipartite graph, we show that the sink-source orientations of G attain extremal values.

2. Bounds of VDB Topological Indices of Digraphs

From now on, when we say that $\phi$ is a symmetric VDB topological index, we mean that $\phi$ is induced by the numbers $\left\{{\phi }_{ij}\right\},$ where $\left(i,j\right)\in K$, and it is defined as in the equivalent definitions (2), (3), or (6). In the first part of this section, we generalize several results of [20] to digraphs.

Let $\phi$ be a symmetric VDB topological index. Consider the function $f_{ij}=\frac{ij{\phi }_{ij}}{i+j}$ defined over the set K. For each $\left(r,s\right)\in K$, consider the subset of K

$$K_{rs}=\left\{\left(i,j\right)\in K:\left(i,j\right)\ne \left(r,s\right)\right\}.$$

Recall that q is the number of vertices which are sink or source vertices of a digraph D.

Lemma 1.

Let φ be a symmetric VDB topological index and $D\in {\mathcal{D}}_n$. Let $\left(r,s\right)\in K$. Then

$$2\phi \left(D\right)=\left(2n-q\right)f_{rs}+\sum _{\left(i,j\right)\in K_{rs}}\left(f_{ij}-f_{rs}\right)\frac{i+j}{ij}{p}_{ij}.$$

Proof. 

The numbers $\left\{p_{ij}\right\}$ defined in (4) in (5) satisfy the relation (see (10) in [1])

$$\sum _{\left(i,j\right)\in K}\left(\frac{1}{i}+\frac{1}{j}\right)p_{ij}=2n-\left(n_0^{+}+n_0^{-}\right).$$

(7)

Note that $q=n_0^{+}+n_0^{-}$. By (7),

$$\frac{r+s}{rs}{p}_{rs}+\sum _{\left(i,j\right)\in K_{rs}}\left(\frac{1}{i}+\frac{1}{j}\right)p_{ij}=2n-q,$$

which implies

$$p_{rs}=\frac{rs}{r+s}\left(2n-q-\sum _{\left(i,j\right)\in K_{rs}}\left(\frac{1}{i}+\frac{1}{j}\right)p_{ij}\right).$$

(8)

On the other hand,

$$\phi \left(D\right)=\frac{1}{2}{p}_{rs}{\phi }_{rs}+\frac{1}{2}\sum _{\left(i,j\right)\in K_{rs}}{p}_{ij}{\phi }_{ij}.$$

(9)

Now, substituting (8) in (9), we deduce

$$\phi \left(D\right)=\frac{1}{2}{f}_{rs}\left(2n-q\right)+\frac{1}{2}\sum _{\left(i,j\right)\in K_{rs}}{p}_{ij}\frac{i+j}{ij}\left(f_{ij}-f_{rs}\right).$$

□

Let $\phi$ be a symmetric VDB topological index with associated function $f_{ij}=\frac{ij{\phi }_{ij}}{i+j}$. Define the sets

$$K_{\min }\left(f\right)=\left\{\left(r,s\right)\in K:f_{rs}=\underset{\left(i,j\right)\in K}{min}{f}_{ij}\right\},$$

and

$$K_{\max }\left(f\right)=\left\{\left(r,s\right)\in K:f_{rs}=\underset{\left(i,j\right)\in K}{max}{f}_{ij}\right\}.$$

We will denote by $K_{min}^c\left(f\right)$ and $K_{max}^c\left(f\right)$ the complements of $K_{min}\left(f\right)$ and $K_{max}\left(f\right)$ in K, respectively. We now generalize ([20], Theorem 2.3) to digraphs.

Theorem 1.

Let φ be a symmetric VDB topological index and $D\in {\mathcal{D}}_n$. Then

$$\frac{1}{2}\left(2n-q\right)\underset{\left(i,j\right)\in K}{min}{f}_{ij}\le \phi \left(D\right)\le \frac{1}{2}\left(2n-q\right)\underset{\left(i,j\right)\in K}{max}{f}_{ij}.$$

Moreover, equality on the left occurs, and only if $p_{xy}=0$ for all $\left(x,y\right)\in K_{\min }^c\left(f\right)$. Equality on the right occurs, and only if $p_{xy}=0$ for all $\left(x,y\right)\in K_{\max }^c\left(f\right)$.

Proof. 

Assume that $f_{rs}={\displaystyle \underset{\left(i,j\right)\in K}{max}}{f}_{ij}$, where $\left(r,s\right)\in K$. By Lemma 1 and the fact that $f_{ij}\le$$f_{rs}$ for all $\left(i,j\right)\in K$, we deduce

$$\begin{array}{ccc}\hfill \phi \left(D\right)& =& \frac{1}{2}\left(\left(2n-q\right)f_{rs}+\sum _{\left(i,j\right)\in K_{rs}}\left(f_{ij}-f_{rs}\right)\frac{i+j}{ij}{p}_{ij}\right)\hfill \\ & \le & \frac{1}{2}\left(2n-q\right)f_{rs}.\hfill \end{array}$$

(10)

On the other hand, since $\left(f_{ij}-f_{rs}\right)\frac{i+j}{ij}{p}_{ij}=0$ for all $\left(i,j\right)\in K_{\max }\left(f\right)$, it is clear that

$$\sum _{\left(i,j\right)\in K_{rs}}\left(f_{ij}-f_{rs}\right)\frac{i+j}{ij}{p}_{ij}=0$$

if, and only if $p_{xy}=0$ for all $\left(x,y\right)\in K_{\max }^c\left(f\right)$. By inequality (10), this is equivalent to $\phi \left(D\right)=\frac{1}{2}\left(2n-n_0^{+}-n_0^{-}\right){\displaystyle \underset{\left(i,j\right)\in K}{max}}{f}_{ij}$. The proof of the left inequality (and the equality condition) is similar. □

So by Theorem 1, in order to find extremal values of a VDB topological index $\phi$ over ${\mathcal{D}}_n$, we must find $K_{\min }\left(f\right)$ and $K_{\max }\left(f\right)$, where $f=\frac{ij{\phi }_{ij}}{i+j}$. Fortunately, these were computed for the main VDB topological indices in [21] (see Table 1).

Table 1. $K_{\min }\left(f\right)$ and $K_{\max }\left(f\right)$ for some VDB topological indices.

VDB Index	Notation	${\mathit{\phi }}_{\mathit{ij}}$	${\mathit{K}}_{min}\left(\mathit{f}\right)$	${\mathit{K}}_{max}\left(\mathit{f}\right)$
First Zagreb [13]	${\mathcal{M}}_1$	$i+j$	$\left(1,1\right)$	$\left(n-1,n-1\right)$
Second Zagreb [13]	${\mathcal{M}}_2$	$ij$	$\left(1,1\right)$	$\left(n-1,n-1\right)$
Randić [14]	$\chi$	$\frac{1}{\sqrt{ij}}$	$\left(1,n-1\right)$	$\left\{\left(i,j\right)\in K:i=j\right\}$
Harmonic [15]	$\mathcal{H}$	$\frac{2}{i+j}$	$\left(1,n-1\right)$	$\left\{\left(i,j\right)\in K:i=j\right\}$
Geometric-Arithmetic [16]	$\mathcal{GA}$	$\frac{2\sqrt{ij}}{i+j}$	$\left(1,n-1\right)$	$\left(n-1,n-1\right)$
Sum-Connectivity [17]	$\mathcal{SC}$	$\frac{1}{\sqrt{i+j}}$	$\left(1,n-1\right)$	$\left(n-1,n-1\right)$
Atom-Bond-Connectivity [18]	$\mathcal{ABC}$	$\sqrt{\frac{i+j-2}{ij}}$	$\left(1,1\right)$	$\left(n-1,n-1\right)$
Augmented Zagreb [19]	$\mathcal{AZ}$	${\left(\frac{ij}{i+j-2}\right)}^3$	$\left(1,n-1\right)$	$\left(n-1,n-1\right)$

An important class of digraphs which occur frequently as extremal values of VDB topological indices are the arc-balanced digraphs, which we define as follows.

Definition 1.

A digraph D is arc-balanced if $d_u^{+}=d_v^{-}$, for every arc $uv$ of D, and $q=0$.

A regular digraph is a digraph D such that $d_u^{+}=d_u^{-}=r$, for all vertices u in D, where r is a positive integer. Clearly, every regular digraph is arc-balanced.

Example 1.

The digraphs in Figure 2 are arc-balanced but not regular digraphs.

Figure 2. Arc-balanced digraphs.

Now we can give sharp upper and lower bounds for all VDB topological indices listed in Table 1. The following result is clear.

Lemma 2.

Let $D\in {\mathcal{D}}_n$.

1.

$p_{ij}=0$ for all $\left(i,j\right) \ne \left(1,1\right)$ if, and only if

$$D=\underset{i=1}{\overset{k_1}{⨁}}{\overrightarrow{P}}_{n_i}\oplus \underset{j=1}{\overset{k_2}{⨁}}{\overrightarrow{C}}_{n_j},$$

for some nonnegative integers $k_1$ and $k_2$.

2.

$p_{ij}=0$ for all $\left(i,j\right) \in K$ such that $i<j$ and $q=0$ ⇔ D is an arc-balanced digraph;

3.

$p_{ij}=0$ for all $\left(i,j\right) \in K$ such that $\left(i,j\right) \ne \left(n-1,n-1\right)$ ⇔ $D=K_n;$

4.

$p_{ij}=0$ for all $\left(i,j\right) \ne \left(1,n-1\right)$ and $q=n\iff D={\overrightarrow{K}}_{1,n-1}$ or $D={\overrightarrow{K}}_{n-1,1};$

5.

$p_{ij}=0$ for all $\left(i,j\right) \ne \left(1,1\right)$ and $q=n\iff n$ is even and $D=\frac{n}{2}{\overrightarrow{P}}_2.$

6.

$p_{ij}=0$ for all $\left(i,j\right) \ne \left(1,1\right)$ and $q=n-1$ ⇔ n is odd and $D=\frac{n-3}{2}{\overrightarrow{P}}_2\oplus {\overrightarrow{P}}_3$.

Lemma 3.

Assume that n is odd. Let $D\in {\mathcal{D}}_n$. If $q=n$, then $p_{11}\le \frac{n-3}{2}$.

Proof. 

Every vertex of D is a sink vertex or a source vertex. Consequently,

$$D=E\oplus p_{11}{\overrightarrow{P}}_2,$$

where $p_{11}\left(E\right) = 0$. In particular,

$$n=n\left(E\right)+2p_{11}.$$

Since n is odd, then $n\left(E\right)$ is also odd. Moreover, $n\left(E\right) \ge 3$, since D has no isolated vertices. Hence,

$$2p_{11}=n-n\left(E\right)\le n-3.$$

□

Corollary 1.

Let $D\in {\mathcal{D}}_n$. Then

1.

$$⌈\frac{n}{2}⌉\le {\mathcal{M}}_1\left(D\right)\le n{\left(n-1\right)}^2.$$

(a) 

Equality on the left occurs ⇔ n is even and $D=\frac{n}{2}{\overrightarrow{P}}_2$ or n is odd, and $D=\frac{n-3}{2}{\overrightarrow{P}}_2\oplus {\overrightarrow{P}}_3$;

(b) 

Equality on the right occurs $\iff D=K_n.$

2.

$$\left.\begin{array}{ccc}\frac{n}{4}& & \mathit{if} n \mathit{even}\\ \frac{n+1}{4}& & \mathit{if} n \mathit{odd}\end{array}\right\}\le {\mathcal{M}}_2\left(D\right)\le \frac{1}{2}n{\left(n-1\right)}^3.$$

(a) 

Equality on the left occurs ⇔ n is even and $D=\frac{n}{2}{\overrightarrow{P}}_2$ or n is odd and $D=\frac{n-3}{2}{\overrightarrow{P}}_2\oplus {\overrightarrow{P}}_3$;

(b) 

Equality on the right occurs $\iff D=K_n.$

3.

$$\frac{1}{2}\sqrt{n-1}\le \chi \left(D\right)\le \frac{n}{2}.$$

(a) 

Equality on the left occurs $\iff D={\overrightarrow{K}}_{1,n-1}$ or $D={\overrightarrow{K}}_{n-1,1};$

(b) 

Equality on the right occurs $\iff D$ is an arc-balanced digraph.

4.

$$\frac{n-1}{n}\le \mathcal{H}\left(D\right)\le \frac{n}{2}.$$

(a) 

Equality on the left occurs $\iff D={\overrightarrow{K}}_{1,n-1}$ or $D={\overrightarrow{K}}_{n-1,1};$

(b) 

Equality on the right occurs $\iff D$ is an arc-balanced digraph.

5.

$$\frac{{\left(n-1\right)}^{\frac{3}{2}}}{n}\le \mathcal{GA}\left(D\right)\le \frac{n}{2\sqrt[3]{{\left(n-1\right)}^4}}.$$

(a) 

Equality on the left occurs $\iff D={\overrightarrow{K}}_{1,n-1}$ or $D={\overrightarrow{K}}_{n-1,1};$

(b) 

Equality on the right occurs $\iff D=K_n.$

6.

$$\frac{n-1}{2\sqrt{n}}\le \mathcal{SC}\left(D\right)\le \frac{1}{4}n\sqrt{2\left(n-1\right)}.$$

(a) 

Equality on the left occurs $\iff D={\overrightarrow{K}}_{1,n-1}$ or $D={\overrightarrow{K}}_{n-1,1};$

(b) 

Equality on the right occurs $\iff D=K_n.$

7.

$$0\le \mathcal{ABC}\left(D\right)\le \frac{n}{2}\sqrt{2\left(n-2\right)}.$$

(a) 

Equality on the left occurs $\iff D={\displaystyle \underset{i=1}{\overset{k_1}{⨁}}}{\overrightarrow{P}}_{n_i}\oplus {\displaystyle \underset{j=1}{\overset{k_2}{⨁}}}{\overrightarrow{C}}_{n_j}$, for some nonnegative integers $k_1,k_2$.

(b) 

Equality on the right occurs $\iff D=K_n.$

8.

$$\frac{1}{2}\frac{{\left(n-1\right)}^4}{{\left(n-2\right)}^3}\le \mathcal{AZ}\left(D\right)\le \frac{1}{16}n\frac{{\left(n-1\right)}^7}{{\left(n-2\right)}^3}.$$

(a) 

Equality on the left occurs $\iff D={\overrightarrow{K}}_{1,n-1}$ or $D={\overrightarrow{K}}_{n-1,1};$

(b) 

Equality on the right occurs $\iff D=K_n.$

Proof. 

Recall that $f_{ij}=\frac{ij{\phi }_{ij}}{i+j}$ is the associated function of the symmetric VDB topological index $\phi$. The expressions for $f_{ij}$ are shown in Table 2.

Table 2. $f_{ij}$ for some VDB topological Indices.

VDB Index	${\mathcal{M}}_1$	${\mathcal{M}}_2$	$\mathit{\chi }$	$\mathcal{H}$	$\mathcal{GA}$	$\mathcal{SC}$	$\mathcal{ABC}$	$\mathcal{AZ}$
$f_{ij}$	$ij$	$\frac{{\left(ij\right)}^2}{i+j}$	$\frac{\sqrt{ij}}{i+j}$	$\frac{2ij}{{\left(i+j\right)}^2}$	$\frac{2{\left(ij\right)}^{\frac{3}{2}}}{{\left(i+j\right)}^2}$	$\frac{ij}{{\left(i+j\right)}^{\frac{3}{2}}}$	$\frac{\sqrt{ij\left(i+j-2\right)}}{i+j}$	$\frac{{\left(ij\right)}^4}{\left(i+j\right){\left(i+j-2\right)}^3}$

Since $0\le q\le n$, we easily deduce the result from Theorem 1 and Lemma 2.

We only have to separately consider ${\mathcal{M}}_1$ and ${\mathcal{M}}_2$ when n is odd. By Theorem 1,

$$2{\mathcal{M}}_1\left(D\right)\ge 2n-q\ge n.$$

(11)

Since n is odd, $2{\mathcal{M}}_1\left(D\right)>n$, and so $2{\mathcal{M}}_1\left(D\right)\ge n+1$. Equivalently,

$${\mathcal{M}}_1\left(D\right)\ge (n+1)/2=\lceil n/2\rceil .$$

For the equality condition, it is clear that ${\mathcal{M}}_1\left(\frac{n-3}{2}{\overrightarrow{P}}_2\oplus {\overrightarrow{P}}_3\right)=\frac{n+1}{2}$. Conversely, suppose that ${\mathcal{M}}_1\left(D\right)=\frac{n+1}{2}$. Then by (11),

$$n+1\ge 2n-q,$$

which implies $q\ge n-1$. So there are only two possibilities: $q=n-1$ and $q=n$. If $q=n$, then by Lemma 3, $p_{11}\le \frac{n-3}{2}$. On the other hand, by Lemma 1 applied to $\left(r,s\right)=\left(1,2\right),$

$$\begin{array}{ccc}\hfill n+1& =& 2{\mathcal{M}}_1\left(D\right)=2n+\sum _{\left(i,j\right)\ne \left(1,2\right)}\left(ij-2\right)\frac{i+j}{ij}{p}_{ij}\hfill \\ & =& 2n+\left(-1\right)2p_{11}+\sum _{\begin{array}{c}\left(i,j\right)\ne \left(1,2\right)\\ \left(i,j\right)\ne \left(1,1\right)\end{array}}\left(ij-2\right)\frac{i+j}{ij}{p}_{ij}.\hfill \end{array}$$

Thus,

$$0\le \sum _{\begin{array}{c}\left(i,j\right)\ne \left(1,2\right)\\ \left(i,j\right)\ne \left(1,1\right)\end{array}}\left(ij-2\right)\frac{i+j}{ij}{p}_{ij}=2p_{11}-n+1,$$

which implies $p_{11}\ge \frac{n-1}{2}$, a contradiction. Hence, $q=n-1$. Consequently,

$${\mathcal{M}}_1\left(D\right)=\frac{n+1}{2}=\frac{1}{2}\left(2n-q\right).$$

It follows from Theorem 1 that $p_{ij}=0$ for all $\left(i,j\right)\ne \left(1,1\right)$. Finally, by Lemma 2,

$$D=\frac{n-3}{2}{\overrightarrow{P}}_2\oplus {\overrightarrow{P}}_3.$$

The case of ${\mathcal{M}}_2$ when n is odd is similar.

In the case of the $\mathcal{ABC}$ index, note that ${\phi }_{ij}=0$ if, and only if $\left(i,j\right)=\left(1,1\right)$. Then it is clear that

$$\mathcal{ABC}\left(\underset{i=1}{\overset{k_1}{⨁}}{\overrightarrow{P}}_{n_i}\oplus \underset{j=1}{\overset{k_2}{⨁}}{\overrightarrow{C}}_{n_j}\right)=0.$$

Conversely, if D is a digraph such that $0=\mathcal{ABC}\left(D\right)$, then

$$0=\mathcal{ABC}\left(D\right)=\frac{1}{2}\sum _{\left(i,j\right)\in K}{p}_{ij}{\phi }_{ij}=\frac{1}{2}\sum _{\begin{array}{c}\left(i,j\right)\in K\\ \left(i,j\right)\ne \left(1,1\right)\end{array}}{p}_{ij}{\phi }_{ij},$$

which implies $p_{ij}=0$ for all $\left(i,j\right)\ne \left(1,1\right)$. Hence, by part 1 of Lemma 2, $D={⨁}_{i=1}^{k_1}{\overrightarrow{P}}_{n_i}\oplus {⨁}_{j=1}^{k_2}{\overrightarrow{C}}_{n_j}$. □

Remark 1.

Using a linear programming modeling technique, the authors in [22] find some of the extremal values given in Corollary 1.

Now we give bounds of VDB topological indices in terms of the number of arcs. Let $\phi$ be a symmetric VDB topological index. Let us define

$$L_{max}=L_{max}\left(\phi \right)=\left\{\left(i,j\right)\in K:{\phi }_{ij}=\underset{K}{max}{\phi }_{ij}\right\},$$

and

$$L_{min}=L_{min}\left(\phi \right)=\left\{\left(i,j\right)\in K:{\phi }_{ij}=\underset{K}{min}{\phi }_{ij}\right\}.$$

The complements in K are denoted by $L_{max}^c$ and $L_{min}^c$, respectively.

Theorem 2.

Let φ be a symmetric VDB topological index. If D is a digraph with a arcs, then

$$\frac{1}{2}a\left(\underset{K}{min}{\phi }_{ij}\right)\le \phi \left(D\right)\le \frac{1}{2}a\left(\underset{K}{max}{\phi }_{ij}\right).$$

Equality on the left occurs if, and only if $p_{ij}=0$ for all $\left(i,j\right)\in L_{min}^c$. Equality on the right occurs if, and only if $p_{ij}=0$ for all $\left(i,j\right)\in L_{max}^c$.

Proof. 

From (2) and (1),

$$\phi \left(D\right)=\frac{1}{2}\sum _K{p}_{ij}{\phi }_{ij}\le \frac{1}{2}\sum _K{p}_{ij}\underset{K}{max}{\phi }_{ij}=\frac{1}{2}a\left(\underset{K}{max}{\phi }_{ij}\right).$$

(12)

If $\phi \left(D\right)=\frac{1}{2}a\left(\underset{K}{max}{\phi }_{ij}\right)$, then by (12)

$$p_{ij}\left({\phi }_{ij}-\underset{K}{max}{\phi }_{ij}\right)=0,$$

for all $\left(i,j\right)\in K$. Hence, if $\left(i,j\right)\in L_{max}^c$, then ${\phi }_{ij}-\underset{K}{max}{\phi }_{ij}\ne 0$ and so $p_{ij}=0$.

Conversely, if $p_{ij}=0$ for all $\left(i,j\right)\in L_{max}^c$, then

$$\begin{array}{ccc}\hfill \phi \left(D\right)& =& \frac{1}{2}\sum _K{p}_{ij}{\phi }_{ij}=\frac{1}{2}\sum _{L_{max}}{p}_{ij}{\phi }_{ij}+\frac{1}{2}\sum _{L_{max}^c}{p}_{ij}{\phi }_{ij}\hfill \\ & =& \frac{1}{2}\sum _{L_{max}}{p}_{ij}{\phi }_{ij}=\frac{1}{2}a\left(\underset{K}{max}{\phi }_{ij}\right).\hfill \end{array}$$

The proof of the left inequality (and equality) is similar. □

3. Bounds of VDB Topological Indices of Tree Orientations

The set of oriented trees with n vertices is denoted by $\mathcal{OT}\left(n\right)$. It is our interest in this section to determine the extremal values of a VDB topological index over $\mathcal{OT}\left(n\right)$. Clearly, $a=n-1$ for every $T\in \mathcal{OT}\left(n\right)$. Hence, by Theorem 2 we deduce the following.

Corollary 2.

Let $T\in \mathcal{OT}\left(n\right)$. Then

$$\frac{1}{2}\left(n-1\right)\underset{K}{min}{\phi }_{ij}\le \phi \left(T\right)\le \frac{1}{2}\left(n-1\right)\underset{K}{max}{\phi }_{ij}.$$

Equality on the left occurs if, and only if $p_{ij}=0$ for all $\left(i,j\right)\in L_{min}^c$. Equality on the right occurs if, and only if $p_{ij}=0$ for all $\left(i,j\right)\in L_{max}^c$.

Now we can obtain a first list of sharp upper and lower bounds for some VDB topological indices over $\mathcal{OT}\left(n\right)$.

Theorem 3.

Let $T\in \mathcal{OT}\left(n\right)$. Then

1.

$\frac{1}{2}\sqrt{n-1}\le \chi \left(T\right)\le \frac{n-1}{2};$

2.

$\frac{n-1}{n}\le \mathcal{H}\left(T\right)\le \frac{n-1}{2};$

3.

$\frac{{\left(n-1\right)}^{\frac{3}{2}}}{n}\le \mathcal{GA}\left(T\right)\le \frac{n-1}{2};$

4.

$\frac{n-1}{2\sqrt{n}}\le \mathcal{SC}\left(T\right)\le \frac{\sqrt{2}}{4}\left(n-1\right);$

5.

$\frac{1}{2}\frac{{\left(n-1\right)}^4}{{\left(n-2\right)}^3}\le \mathcal{AZ}\left(T\right)$.

Moreover, equality on the left of 1–5 occurs $\iff T={\overrightarrow{K}}_{1,n-1}$ or $T={\overrightarrow{K}}_{n-1,1}$. Equality on the right of 1–4 occurs $\iff T={\overrightarrow{P}}_n$.

Proof. 

The inequalities on the left (and equality conditions) are immediate consequence of Corollary 1. The inequalities on the right of 1–4 are consequence of Corollary 2 having in mind Table 3.

Table 3. $L_{max}$ and ${\displaystyle \underset{K}{max}}{\phi }_{ij}$ for $\chi$, $\mathcal{H}$, $\mathcal{GA}$, and $\mathcal{SC}$.

VDB Index	${\mathit{\phi }}_{\mathit{ij}}$	${\mathit{L}}_{max}$	${\displaystyle \underset{\mathit{K}}{max}}{\mathit{\phi }}_{\mathit{ij}}$
$\chi$	$\frac{1}{\sqrt{ij}}$	$\left(1,1\right)$	1
$\mathcal{H}$	$\frac{2}{i+j}$	$\left(1,1\right)$	1
$\mathcal{GA}$	$\frac{2\sqrt{ij}}{i+j}$	$\left\{\left(i,j\right)\in K:i=j\right\}$	1
$\mathcal{SC}$	$\frac{1}{\sqrt{i+j}}$	$\left(1,1\right)$	$\frac{1}{\sqrt{2}}$

We also use the fact that $T\in \mathcal{OT}\left(n\right)$ is such that $p_{ij}=0$ for all $\left(i,j\right)\ne \left(1,1\right)$ if, and only if $T={\overrightarrow{P}}_n$. Similarly, $p_{ij}=0$ for all $\left(i,j\right)$ such that $i<j$ if, and only if $T={\overrightarrow{P}}_n$. □

Theorem 4.

Let $T\in \mathcal{OT}\left(n\right)$. Then

1.

$0\le \mathcal{ABC}\left(T\right)\le \frac{1}{2}\sqrt{\left(n-1\right)\left(n-2\right)};$

2.

$\left(n-1\right)\le {\mathcal{M}}_1\left(T\right);$

3.

$\frac{1}{2}\left(n-1\right)\le {\mathcal{M}}_2\left(T\right).$

Moreover, equality on the left of 1–3 occurs $\iff T={\overrightarrow{P}}_n$. Equality on the right of 1 occurs $\iff T={\overrightarrow{K}}_{1,n-1}$ or $T={\overrightarrow{K}}_{n-1,1}.$

Proof. 

The inequalities on the left of 1–3 (and equality conditions) are a consequence of Corollary 2, having in mind Table 4.

Table 4. $L_{min}$ and ${\displaystyle \underset{K}{min}}{\phi }_{ij}$ for $\mathcal{ABC}$, ${\mathcal{M}}_1$, and ${\mathcal{M}}_2$.

VDB Index	${\mathit{\phi }}_{\mathit{ij}}$	${\mathit{L}}_{min}$	${\displaystyle \underset{\mathit{K}}{min}}{\mathit{\phi }}_{\mathit{ij}}$
$\mathcal{ABC}$	$\sqrt{\frac{i+j-2}{ij}}$	$\left(1,1\right)$	0
${\mathcal{M}}_1$	$i+j$	$\left(1,1\right)$	2
${\mathcal{M}}_2$	$ij$	$\left(1,1\right)$	1

And the fact that $T\in \mathcal{OT}\left(n\right)$ is such that $p_{ij}=0$ for all $\left(i,j\right)\ne \left(1,1\right)$ if, and only if $T={\overrightarrow{P}}_n$. On the other hand, the right inequality in 1 holds again by Corollary 2, bearing in mind Table 5.

Table 5. $L_{max}$ and ${\displaystyle \underset{K}{max}}{\phi }_{ij}$ for $\mathcal{ABC}$.

VDB Index	${\mathit{\phi }}_{\mathit{ij}}$	${\mathit{L}}_{max}$	${\displaystyle \underset{\mathit{K}}{max}}{\mathit{\phi }}_{\mathit{ij}}$
$\mathcal{ABC}$	$\sqrt{\frac{i+j-2}{ij}}$	$\left(1,n-1\right)$	$\sqrt{\frac{n-2}{n-1}}$

And the fact that $T\in \mathcal{OT}\left(n\right)$ is such that $p_{ij}=0$ for all $\left(i,j\right)\ne \left(1,n-1\right)$ if, and only if $T={\overrightarrow{K}}_{1,n-1}$ or $T={\overrightarrow{K}}_{n-1,1}.$ □

The only extremal values we have not determined yet are the maximal values of ${\mathcal{M}}_1,{\mathcal{M}}_2,$ and $\mathcal{AZ}$ over $\mathcal{OT}\left(n\right)$. The problem in these indices is that $L_{max}=\left(n-1,n-1\right)$, and there is no oriented tree such that $p_{ij}=0$ for all $\left(i,j\right)\ne \left(n-1,n-1\right)$. In the next section we will show that the maximum value of ${\mathcal{M}}_1$ and ${\mathcal{M}}_2$ over $\mathcal{OT}\left(n\right)$ is attained in ${\overrightarrow{K}}_{1,n-1}$ or ${\overrightarrow{K}}_{n-1,1}$ (see Theorem 6). We propose the following problem.

Problem 1.

Find the maximum value of $\mathcal{AZ}$ over $\mathcal{OT}\left(n\right)$.

4. Bounds of VDB Topological Indices over Orientations of a Fixed Graph

Let $\phi$ be a symmetric VDB topological index and G a graph. Let $\mathcal{O}\left(G\right)$ be the set of orientations of the graph G. Our main concern now is to determine the extremal values of a symmetric VDB topological index over $\mathcal{O}\left(G\right)$. In order to do this, let us define a partial order over K as follows: if $\left(i,j\right),\left(k,l\right)\in K$, then

$$\left(i,j\right)⪯\left(k,l\right)\iff i\le k\mathrm{and}j\le l.$$

Definition 2.

Let φ be a symmetric VDB topological index. We say that φ is nondecreasing (resp. nonincreasing) over K, if for every $\left(i,j\right),\left(k,l\right)\in K:$

$$\left(i,j\right)⪯\left(k,l\right)\Rightarrow {\phi }_{ij}\le {\phi }_{kl}(\mathrm{resp}.{\phi }_{ij}\ge {\phi }_{kl}).$$

Furthermore, if for every $\left(i,j\right),\left(k,l\right)\in K:$

$$\left(i,j\right)⪯\left(k,l\right)\mathrm{and}{\phi }_{ij}={\phi }_{kl}\Rightarrow \left(i,j\right)=\left(k,l\right),$$

we will say that φ is strictly nondecreasing (resp. strictly nonincreasing).

Example 2.

Consider the generalized Randić index ${\chi }_{\alpha }$ induced by the numbers ${\left(ij\right)}^{\alpha }$, where $\alpha \in \mathbb{R}$, $\alpha \ne 0$. Clearly, ${\chi }_{\alpha }$ is strictly nondecreasing when $\alpha >0$, and strictly nonincreasing when $\alpha <0$. In particular, the Randić index χ is strictly nonincreasing and the second Zagreb index ${\mathcal{M}}_2$ is strictly nondecreasing. Additionally, the harmonic index and the sum-connectivity index are strictly nonincreasing, and the first Zagreb ${\mathcal{M}}_1$ is strictly nondecreasing.

Theorem 5.

Let φ be a strictly nondecreasing (resp. nonincreasing) symmetric VDB topological index and G a graph. Let D be any orientation of G. Then

$$\phi \left(D\right)\le \frac{1}{2}\phi \left(G\right)(\mathrm{resp}.\phi \left(D\right)\ge \frac{1}{2}\phi \left(G\right)).$$

Equality holds if, and only if D is a sink-source orientation of G.

Proof. 

We will assume that $\phi$ is strictly nondecreasing, and the other case is similar. Note that

$$d_u=d_u^{+}+d_u^{-}$$

(13)

for every vertex u of G. Hence, for any arc $uv$ of D, $\left(d_u^{+},d_v^{-}\right)⪯\left(d_u,d_v\right)$. It follows by the nondecreasing property of $\phi$ and (3),

$$\phi \left(D\right)=\frac{1}{2}\sum _{uv\in A}{\phi }_{d_u^{+}{d}_v^{-}}\le \frac{1}{2}\sum _{uv\in G}{\phi }_{d_u{d}_v}=\frac{1}{2}\phi \left(G\right).$$

(14)

If D is a sink-source orientation of G, then $d_u^{+}=0$ or $d_u^{-}=0$, for all vertices u of V. If $vw$ is an arc of D then $d_v^{+}\ne 0$ and $d_w^{-}\ne 0$. Hence, $d_v^{-}=0$ and $d_w^{+}=0$, which implies by (13) that $d_v=d_v^{+}$ and $d_w=d_w^{-}$. Hence,

$$\phi \left(D\right)=\frac{1}{2}\sum _{uv\in A}{\phi }_{d_u^{+}{d}_v^{-}}=\frac{1}{2}\sum _{uv\in G}{\phi }_{d_u{d}_v}=\frac{1}{2}\phi \left(G\right).$$

Conversely, assume that $\phi \left(D\right)=\frac{1}{2}\phi \left(G\right)$. Then by (14), for every $uv\in A$

$$\left(d_u^{+},d_v^{-}\right)⪯\left(d_u,d_v\right)\mathrm{and}{\phi }_{d_u^{+}{d}_v^{-}}={\phi }_{d_u{d}_v}.$$

Now since $\phi$ is strictly nondecreasing, $\left(d_u^{+},d_v^{-}\right)=\left(d_u,d_v\right)$ for every $uv\in A$. Finally, by (13), $d_u^{-}=0$ and $d_v^{+}=0$. This clearly implies that D is a sink-source orientation of G. □

Corollary 3.

Let φ be a strictly nondecreasing (resp. nonincreasing) symmetric VDB topological index and G a bipartite graph. Then the maximal (resp. minimal) value of φ over $\mathcal{O}\left(G\right)$ is attained in a sink-source orientation of G.

Proof. 

We assume that $\phi$ is strictly nondecreasing, and the other case is similar. Since G is a bipartite graph, G has a sink-source orientation which we call E [23]. Let D be any orientation of G. Then by Theorem 5,

$$\phi \left(E\right)=\frac{1}{2}\phi \left(G\right)\ge \phi \left(D\right).$$

□

Example 3.

Consider the path tree $P_n$. By Example 2 and Corollary 3, the sink-source orientation $E\in \mathcal{O}\left(P_n\right)$ depicted in Figure 3 attains the maximal value for ${\mathcal{M}}_1$, ${\mathcal{M}}_2$ and ${\chi }_{\alpha }$ when $\alpha >0$, over $\mathcal{O}\left(P_n\right)$. On the other hand, E attains the minimal value of $\mathcal{H},\mathcal{SC}$ and ${\chi }_{\alpha }$ when $\alpha <0$, over $\mathcal{O}\left(P_n\right).$

Figure 3. Sink-source orientations of $P_n$.

Example 4.

In [24] the authors studied the extreme values of χ on the set of all the orientations of hexagonal chains with k hexagons.

Theorem 6.

Let $T\in \mathcal{OT}\left(n\right)$. Then

1.

${\mathcal{M}}_1\left(T\right)⩽\frac{1}{2}n(n-1)$;

2.

${\mathcal{M}}_2\left(T\right)⩽\frac{1}{2}{(n-1)}^2$.

Moreover, equalities 1–2 occur $\iff T={\overrightarrow{K}}_{1,n-1}$ or $T={\overrightarrow{K}}_{n-1,1}.$

Proof. 

Let G be a tree of order n. If G is different from $S_n$, then [25]

$$\begin{array}{cc}\hfill {\mathcal{M}}_1\left(G\right)& <{\mathcal{M}}_1\left(S_n\right)=n(n-1)\hfill \\ \hfill {\mathcal{M}}_2\left(G\right)& <{\mathcal{M}}_2\left(S_n\right)={(n-1)}^2\hfill \end{array}.$$

Let $T\in \mathcal{OT}\left(n\right)$ and suppose that T is an orientation of a tree G. By Theorem 5 and the above equation,

$$\begin{array}{cc}\hfill {\mathcal{M}}_1\left(T\right)& ⩽\frac{1}{2}{\mathcal{M}}_1\left(G\right)⩽\frac{1}{2}n(n-1)\hfill \\ \hfill {\mathcal{M}}_2\left(T\right)& ⩽\frac{1}{2}{\mathcal{M}}_2\left(G\right)⩽\frac{1}{2}{(n-1)}^2.\hfill \end{array}$$

Equality occurs if, and only if T is a sink-source orientation of $S_n$, in other words, $T={\overrightarrow{K}}_{1,n-1}$ or $T={\overrightarrow{K}}_{n-1,1}$. □