Extremal Graphs to Vertex Degree Function Index for Convex Functions

Abstract

The vertex-degree function index $H_f\left(\mathsf{\Gamma }\right)$ is defined as $H_f\left(\mathsf{\Gamma }\right)={\sum }_{v\in V\left(\mathsf{\Gamma }\right)}f\left(d\left(v\right)\right)$ for a function $f\left(x\right)$ defined on non-negative real numbers. In this paper, we determine the extremal graphs with the maximum (minimum) vertex degree function index in the set of all n-vertex chemical trees, trees, and connected graphs. We also present the Nordhaus–Gaddum-type results for $H_f\left(\mathsf{\Gamma }\right)+H_f\left(\overline{\mathsf{\Gamma }}\right)$ and $H_f\left(\mathsf{\Gamma }\right)·H_f\left(\overline{\mathsf{\Gamma }}\right)$.

1. Introduction

In this paper, the graphs we discuss are simple graphs without multiple edges and loops. The vertex and edge set of $\mathsf{\Gamma }$ will be denoted by $V\left(\mathsf{\Gamma }\right)$ and $E\left(\mathsf{\Gamma }\right)$, and the order and size of $\mathsf{\Gamma }$ will usually be denoted by n and m, respectively. Let a vertex $v\in V\left(\mathsf{\Gamma }\right)$; we denote the degree of v by $d_{\mathsf{\Gamma }}\left(v\right)$ in $\mathsf{\Gamma }$. The neighbors of v in $V\left(\mathsf{\Gamma }\right)$ are denoted by $N_{\mathsf{\Gamma }}\left(v\right)$. For a graph $\mathsf{\Gamma }$, we denote the maximum and minimum degree of $\mathsf{\Gamma }$ by $\Delta \left(\mathsf{\Gamma }\right)$ and $\delta \left(\mathsf{\Gamma }\right)$, respectively. A leaf$v\in V\left(\mathsf{\Gamma }\right)$ is a vertex v satisfied $d_{\mathsf{\Gamma }}\left(v\right)=1$. We call a connected graph without a cycle a tree, denoted by T. A tree whose maximum degree is no more than 4 is called a chemical tree. The star graph with order n, denoted by $S_n$, is a tree with one center vertex and $n-1$ leaves. The disjoint union of two vertex-disjoint graphs ${\mathsf{\Gamma }}_1$ and ${\mathsf{\Gamma }}_2$ will be denoted by ${\mathsf{\Gamma }}_1\cup {\mathsf{\Gamma }}_2$, whose vertex and edge sets are $V\left({\mathsf{\Gamma }}_1\right)\cup V\left({\mathsf{\Gamma }}_2\right)$ and $E\left({\mathsf{\Gamma }}_1\right)\cup E\left({\mathsf{\Gamma }}_2\right)$, respectively. We denote the union of k copies of a graph $\mathsf{\Gamma }$ by $k\mathsf{\Gamma }$. The join of ${\mathsf{\Gamma }}_1$ and ${\mathsf{\Gamma }}_2$ is obtained by joining edges between each vertex of ${\mathsf{\Gamma }}_1$ and all vertices of ${\mathsf{\Gamma }}_2$, denoted by ${\mathsf{\Gamma }}_1\vee {\mathsf{\Gamma }}_2$. For a graph $\mathsf{\Gamma }$, the edge $uv\in E\left(\mathsf{\Gamma }\right)$ and the vertex $w\in V\left(\mathsf{\Gamma }\right)$, $\mathsf{\Gamma }-uv$ mean removing $uv$ from $\mathsf{\Gamma }$ and $\mathsf{\Gamma }-w$, which means removing w from $\mathsf{\Gamma }$ .

A universal vertex of $\mathsf{\Gamma }$ with order n is a vertex v that have $d\left(v\right)=n-1$. An $(n,m)$-graph is the graph with n vertices and m edges. We denote by $\mathsf{\Gamma }(n,m)$ the set of $(n,m)$-graphs. The cyclomatic number of a graph $\mathsf{\Gamma }$ is the minimum number of edges whose deletion transforms $\mathsf{\Gamma }$ into an acyclic graph, denoted by $\gamma \left(\mathsf{\Gamma }\right)$. The set of graphs with order n and cyclomatic number $\gamma$ is denoted by ${\mathsf{\Gamma }}_{n,\gamma }$. We have $\gamma \left(\mathsf{\Gamma }\right)=m-n+1$ for a connected graph $\mathsf{\Gamma }\in {\mathsf{\Gamma }}_{n,\gamma }$.

The vertex-degree function index $H_f\left(\mathsf{\Gamma }\right)$ is denoted by

$$H_f\left(\mathsf{\Gamma }\right)=\sum _{v\in V\left(\mathsf{\Gamma }\right)}f\left(d\left(v\right)\right)$$

for a function $f\left(x\right)$ defined on non-negative real numbers in [1]. For example, the first Zagreb index [2] is defined as $M_1\left(\mathsf{\Gamma }\right)={\sum }_{v\in V\left(\mathsf{\Gamma }\right)}d{\left(v\right)}^2$ when $f\left(x\right)=x^2$, and the forgotten topological index [3] is defined as $F\left(\mathsf{\Gamma }\right)={\sum }_{v\in V\left(\mathsf{\Gamma }\right)}d{\left(v\right)}^3$ when $f\left(x\right)=x^3$. The general first Zagreb index, denoted by ${}^0{R}_{\alpha }\left(\mathsf{\Gamma }\right)$, was defined in [4,5] as ${}^0{R}_{\alpha }\left(\mathsf{\Gamma }\right)={\sum }_{v\in V\left(\mathsf{\Gamma }\right)}d{\left(v\right)}^{\alpha }$, where $\alpha$ is a real number, $\alpha \notin \{0,1\}$. For the mathematical properties of the above topological indices, see [6,7,8,9,10,11] and the references therein. Let v be a leaf of $S_n$, where $n\ge 3$. For $0\le \gamma \le n-2$, the graph obtained from $S_n$ by joining edges v with $\gamma$ other pendant vertices is denoted by $H_{n,\gamma }$ in [12]. Deng [10] obtained the bounds of the Zagreb indices for trees, unicyclic graphs, and bicyclic graphs. Hu and Li determined the connected $(n,m)$-graphs with the minimum and maximum zeroth-order general Randić index in [13]. Li and Zheng [5] obtained a unified approach to the extremal trees for different indices. Some extremal results concerning the general zeroth-order Randić index were deduced in [14,15,16]; also see the survey [12].

In [17], Tomescu obtained that the function $f\left(x\right)$ has property $(P_{\nearrow };P_{\searrow })$ if $\phi (i+1)>\phi \left(i\right);\phi (i+1)<\phi \left(i\right)$, respectively, for every integer $i\ge 0$, where $\phi \left(x\right)=f(x+2)+f\left(x\right)-2f(x+1)$, and he obtained the maximum (minimum) vertex degree function index $H_f\left(\mathsf{\Gamma }\right)$ in the set of all n-vertex connected graphs that have the cyclomatic number $\gamma$ when $0\le \gamma \le n-2$ if $f\left(x\right)$ is strictly convex (concave) and satisfies the property $P_{\nearrow };P_{\searrow }$. Tomescu [18] obtained some structural properties of connected $(n,m)$-graphs which are maximum (minimum) with respect to vertex-degree function index $H_f\left(\mathsf{\Gamma }\right)$, when $f\left(x\right)$ is a strictly convex (concave) function. In the same paper, it is also shown that the unique graph obtained from the star $S_n$ by adding $\gamma$ edges between a fixed pendant vertex v and $\gamma$ other pendant vertices has the maximum general zeroth-order Randić index ${}^0{R}_{\alpha }$ in the set of all n-vertex connected graphs that have the cyclomatic number $\gamma$ when $1\le \gamma \le n-2$ and $\alpha \ge 2$.

Tomescu obtained the following results.

Theorem 1

([18]). In the set of connected $(n,m)$-graphs Γ that have $m\ge n$, the graph that maximizes (minimizes) $H_f\left(\mathsf{\Gamma }\right)$ where $f\left(x\right)$ is strictly convex (concave) possesses the following properties:

(1) 

Γ has a universal vertex v;

(2) 

The subgraph $\mathsf{\Gamma }-v$ consists of some isolated vertices and a nontrivial connected component C, which is maximum (minimum) relatively to $H_g$, where $g\left(x\right)=f(x+1)$. C also contains a universal vertex and no induced subgraph isomorphic to $P_4$ or $C_p$, where $p\ge 4$.

Theorem 2

([17]). If $n\ge 3,1\le \gamma \le n-2$, $f\left(x\right)$ is strictly convex and has property $P_{\nearrow }$, and Γ is a connected n-vertex graph with cyclomatic number γ, then

$$H_f\left(\mathsf{\Gamma }\right)\le f(n-1)+f(\gamma +1)+\gamma f\left(2\right)+(n-\gamma -2)f\left(1\right),$$

with equality if and only if $\mathsf{\Gamma }\cong H_{n,\gamma }\cong K_1\vee (K_{1,\gamma }\cup (n-\gamma -2)K_1)$.

In Section 2, we give upper and lower bounds for the vertex degree function index of connected graphs if $f\left(x\right)$ is a convex and increasing function that has property $P_{\nearrow }$. We obtain sharp upper and lower bounds for the vertex degree function index of trees and chemical trees if $f\left(x\right)$ is a convex and increasing function.

Let $f\left(\mathsf{\Gamma }\right)$ be a graph invariant and n be a positive integer. The Nordhaus–Gaddum Problem is to determine sharp bounds for $f\left(\mathsf{\Gamma }\right)+f\left(\overline{\mathsf{\Gamma }}\right)$ and $f\left(\mathsf{\Gamma }\right)·f\left(\overline{\mathsf{\Gamma }}\right)$ as $\mathsf{\Gamma }$ ranges over the class of all graphs of order n, and to characterize the extremal graphs, i.e., graphs that achieve the bounds. Nordhaus–Gaddum-type relations have received wide attention; see the recent survey [19] by Aouchiche and Hansen and the book chapter by Mao [20].

Denote by $\mathcal{G}\left(n\right)$ the class of connected graphs of order n whose complements are also connected. In Section 3, the upper and lower bounds for $H_f\left(\mathsf{\Gamma }\right)+H_f\left(\overline{\mathsf{\Gamma }}\right)$ and $H_f\left(\mathsf{\Gamma }\right)·H_f\left(\overline{\mathsf{\Gamma }}\right)$ are given for $\mathsf{\Gamma }\in \mathcal{G}\left(n\right)$.

2. Bounds on ${\mathit{H}}_{\mathit{f}}\left(\mathsf{\Gamma }\right)$

At first, we give the following upper bound for $H_f\left(\mathsf{\Gamma }\right)$.

Theorem 3.

Let Γ be an n-vertex $(n\ge 5)$, m-edge graph with a cyclomatic number γ such that $\gamma \in \left[2n-t-3,3n-2t-7\right]$, where $t(1\le t\le n-4)$ is the number of pendant vertices in Γ. If $f\left(x\right)$ is a strictly convex function that has property $P_{\nearrow }$, then

$$\begin{array}{cc}{H}_f\left(\mathsf{\Gamma }\right)\le & f(n-1)+tf\left(1\right)+f(n-t)+f(\gamma -n+t+5)+(\gamma -n+t+4)f\left(3\right)\\ & +(2n-2t-\gamma -5)f\left(2\right)\end{array}$$

with equality if and only if $\mathsf{\Gamma }\cong K_1\vee \left(\left(K_1\vee \left(K_{1,{\gamma }_2}\cup (n_2-{\gamma }_2-2)K_1\right)\right)\cup t{K}_1\right)$, where $n_2=n-t-1,{\gamma }_2=\gamma -n+t+2$.

Proof. 

Let $\mathsf{\Gamma }\in {\mathsf{\Gamma }}_{n,\gamma }$ such that $H_f\left(\mathsf{\Gamma }\right)$ is maximum. By $\left(1\right)$ of Theorem 1, a universal vertex $v_1\in V\left(\mathsf{\Gamma }\right)$ exists, and hence

$$H_f\left(\mathsf{\Gamma }\right)=f(n-1)+H_{g_1}(\mathsf{\Gamma }-v_1),$$

where $g_1\left(x\right)=f(x+1)$. By $\left(2\right)$ of Theorem 1, $\mathsf{\Gamma }-v_1$ consists of some isolated vertices and a nontrivial connected component C. Let ${\mathsf{\Gamma }}_1=\mathsf{\Gamma }-v_1$. Note that t is the number of isolated vertices of $\mathsf{\Gamma }-v_1$; we have

$$H_{g_1}(\mathsf{\Gamma }-v_1)=t{g}_1\left(0\right)+H_{g_1}\left(C\right).$$

Suppose that $m_1,n_1,{\gamma }_1$ and $m_2,n_2,{\gamma }_2$ are the number edges, vertices, and cyclomatic number of ${\mathsf{\Gamma }}_1$, ${\mathsf{\Gamma }}_2$, respectively, where ${\mathsf{\Gamma }}_1=\mathsf{\Gamma }-v_1$ and ${\mathsf{\Gamma }}_2=C$. Since ${\gamma }_2=m_2-n_2+1,m_2=m_1-n_1+1,n_2=n_1-t,m_1=m-n+1,n_1=n-1$, we have $m_2=m-2n+3$ and $n_2=n-t-1$, it follows that ${\gamma }_2=\gamma -2n+t+4$; note that $\gamma \in \left[2n-t-3,3n-2t-7\right]$, so $1\le {\gamma }_2\le n_2-2$ and $n_2\ge 3$, which implies $m_2\ge n_2$. Then, we know that ${\mathsf{\Gamma }}_2$ is a connected $n_2$-vertex graph with cyclomatic number ${\gamma }_2$ and $1\le {\gamma }_2\le n_2-2$, $n_2\ge 3$. So, we can apply Theorem 2 for ${\mathsf{\Gamma }}_2$ and we have

$$H_{g_1}\left({\mathsf{\Gamma }}_2\right)\le g_1(n_2-1)+g_1({\gamma }_2+1)+{\gamma }_2{g}_1\left(2\right)+(n_2-{\gamma }_2-2)g_1\left(1\right)$$

with equality if only if ${\mathsf{\Gamma }}_2\cong K_1\vee \left(K_{1,{\gamma }_2}\cup (n_2-{\gamma }_2-2)K_1\right)$.

Hence, we have

$$\begin{array}{cc}\hfill H_f\left(\mathsf{\Gamma }\right)=& f(n-1)+t{g}_1\left(0\right)+H_{g_1}\left({\mathsf{\Gamma }}_2\right)\hfill \\ \hfill \le & f(n-1)+t{g}_1\left(0\right)+g_1(n_2-1)+g_1({\gamma }_2+1)+{\gamma }_2{g}_1\left(2\right)+(n_2-{\gamma }_2-2)g_1\left(1\right)\hfill \\ \hfill =& f(n-1)+tf\left(1\right)+f(n-t)+f(\gamma -n+t+5)+(\gamma -n+t+4)f\left(3\right)\hfill \\ \hfill & +(2n-2t-\gamma -5)f\left(2\right)\end{array}$$

with equality if only if $\mathsf{\Gamma }\cong K_1\vee \left(\left(K_1\vee \left(K_{1,{\gamma }_2}\cup (n_2-{\gamma }_2-2)K_1\right)\right)\cup t{K}_1\right)$, where $n_2=n-t-1,{\gamma }_2=\gamma -n+t+2$. □

A similar result holds for strictly concave functions $f\left(x\right)$, which have property $P_{\searrow }$: the minimum of $H_f\left(\mathsf{\Gamma }\right)$ is reached in ${\mathsf{\Gamma }}_{n,\gamma }$ if and only if $\mathsf{\Gamma }\cong K_1\vee \left(\left(K_1\vee \left(K_{1,{\gamma }_2}\cup (n_2-{\gamma }_2-2)K_1\right)\right)\cup t{K}_1\right)$, where $n_2=n-t-1,{\gamma }_2=\gamma -n+t+2$.

Lemma 1.

If $f\left(x\right)$ is a convex function, then $f\left(x\right)-f(x-a)\ge f(x-b)-f(x-b-a)$ with equality if and only if $b=0$, where $a,b\ge 0$.

Proof. 

Let $h\left(x\right)=f\left(x\right)-f(x-a)$. Since $f\left(x\right)$ is a convex function, it follows that $f^{\prime }\left(x\right)$ is an increasing function and $h^{\prime }\left(x\right)=f^{\prime }\left(x\right)-f^{\prime }(x-a)\ge 0$. So, $h\left(x\right)$ is an increasing function and $h\left(x\right)\ge h(x-b)$ with equality if and only if $b=0$, and therefore $f\left(x\right)-f(x-a)\ge f(x-b)-f(x-b-a)$. □

We now give a lower bound for $H_f\left(T\right)$.

Theorem 4.

Let T be a tree of order $n(n\ge 4)$. If $f\left(x\right)$ is a convex function, then $H_f\left(T\right)\ge (n-2)f\left(2\right)+2f\left(1\right)$ with equality if and only if $T\cong P_n$.

Proof. 

If $n=4$, then $T\cong S_4$ or $T\cong P_4$. One can easily check that

$$H_f\left(S_4\right)=3f\left(1\right)+f\left(3\right)>2f\left(2\right)+2f\left(1\right)=H_f\left(P_4\right)$$

as $f\left(3\right)-f\left(2\right)>f\left(2\right)-f\left(1\right)$, by Lemma 1. The result holds for $n=4$.

We now suppose that $n\ge 5$. We prove this result by the induction on n. Assume that the result holds for $n-1$ and prove it for n. Let $T^{\prime }$ be a tree of order $n-1$ such that $T-v_j=T^{\prime }$, where $d_T\left(v_j\right)=1$, $v_i=N_T\left(v_j\right)$ and $d_{T^{\prime }}\left(v_i\right)=d_T\left(v_i\right)-1=p-1$. Thus, we have $H_f\left(T^{\prime }\right)\ge (n-3)f\left(2\right)+2f\left(1\right)$ with equality if and only if $T^{\prime }\cong P_{n-1}$. One can easily see that

$$H_f\left(T\right)=H_f\left(T^{\prime }\right)+f\left(p\right)-f(p-1)+f\left(1\right).$$

Since $f\left(x\right)$ is a convex function, it follows from Lemma 1 that $f\left(p\right)-f(p-1)\ge f\left(2\right)-f\left(1\right)$ with equality if and only if $p=2$. Therefore, by the induction hypothesis with the above results, we obtain

$$\begin{array}{cc}\hfill H_f\left(T\right)& =H_f\left(T^{\prime }\right)+f\left(p\right)-f(p-1)+f\left(1\right)\hfill \\ \hfill & \ge (n-3)f\left(2\right)+3f\left(1\right)+f\left(p\right)-f(p-1)\hfill \\ \hfill & \ge (n-2)f\left(1\right)+2f\left(1\right)\hfill \end{array}$$

and the result holds by induction. Moreover, the equality holds if and only if $T^{\prime }\cong P_{n-1}$ and $d_T\left(v_i\right)=p=2$, that is, if and only if $T\cong P_n$. □

Corollary 1.

Let T be a chemical tree of order n $(n\ge 4)$. If $f\left(x\right)$ is a convex function, then $H_f\left(T\right)\ge (n-2)f\left(2\right)+2f\left(1\right)$ with equality if and only if $T\cong P_n$.

Using Theorem 4, we obtain a lower bound for $H_f\left(\mathsf{\Gamma }\right)$.

Theorem 5.

Let Γ be a connected graph of order $n(n\ge 4)$. If $f\left(x\right)$ is a convex and increasing function, then $H_f\left(\mathsf{\Gamma }\right)\ge (n-2)f\left(2\right)+2f\left(1\right)$ with equality if and only if $\mathsf{\Gamma }\cong P_n$.

Proof. 

Since $f\left(x\right)$ is an increasing function, it follows that $f(x+1)+f(y+1)\ge f\left(x\right)+f\left(y\right)$, and hence $H_f(\mathsf{\Gamma }+e)\ge H_f\left(\mathsf{\Gamma }\right)$, where e is an edge joining between two non-adjacent vertices in $\mathsf{\Gamma }$. Clearly, for the graph $\mathsf{\Gamma }$ of order n, we have $H_f\left(\mathsf{\Gamma }\right)\ge H_f\left(T\right)$, where T is a tree of order n. This result with Theorem 4, we obtain $H_f\left(\mathsf{\Gamma }\right)\ge H_f\left(T\right)\ge (n-2)f\left(2\right)+2f\left(1\right)$. Moreover, the equality holds if and only if $T\cong P_n$. □

A complete split graph $CS(n,\alpha )$ is defined as the graph join ${\overline{K}}_{\alpha }\vee K_{n-\alpha }$, where $\alpha$ is the independence number of graph $CS(n,\alpha )$.

Theorem 6.

Let Γ be a connected graph of order $n(n\ge 4)$ with independence number α. If $f\left(x\right)$ is a strictly increasing function, then $H_f\left(\mathsf{\Gamma }\right)\le (n-\alpha )f(n-1)+\alpha f(n-\alpha )$ with equality if and only if $\mathsf{\Gamma }\cong CS(n,\alpha )$.

Proof. 

Since $f\left(x\right)$ is a strictly increasing function, it follows that $f(x+1)+f(y+1)>f\left(x\right)+f\left(y\right)$, and hence $H_f(\mathsf{\Gamma }+e)>H_f\left(\mathsf{\Gamma }\right)$, where e is an edge joining between two non-adjacent vertices in $\mathsf{\Gamma }$. Since $\mathsf{\Gamma }$ is a graph of order n with independence number $\alpha$, we must have that $\mathsf{\Gamma }$ is a subgraph of $CS(n,\alpha )$. If $\mathsf{\Gamma }\cong CS(n,\alpha )$, then $H_f\left(\mathsf{\Gamma }\right)=(n-\alpha )f(n-1)+\alpha f(n-\alpha )$; hence, the equality holds. Otherwise, $\mathsf{\Gamma }\ncong CS(n,\alpha )$. Since $\mathsf{\Gamma }$ is a subgraph of $CS(n,\alpha )$ and $H_f(\mathsf{\Gamma }+e)>H_f\left(\mathsf{\Gamma }\right)$, we obtain $H_f\left(\mathsf{\Gamma }\right)<H_f(\mathsf{\Gamma }+e)<\cdots <H_f(CS(n,\alpha )-e_1)$ $<H_f\left(CS(n,\alpha )\right)=(n-\alpha )f(n-1)+$$\alpha f(n-\alpha ),$ where $e_1$ is an edge in $CS(n,\alpha )$. This completes the proof of the theorem. □

Let C be the set of pendant vertices, and let A be the set of non-leaf vertices that have at least 2 neighbor vertices, each of which are not leaves. Let B be the set of non-leaf vertices that have only one neighbor vertex, which is not a leaf. Note that $V\left(\mathsf{\Gamma }\right)=A\cup B\cup C$.

Lemma 2.

Let Γ be a graph of order n, and $f\left(x\right)$ be a convex function.

(1) 

If $u\in A$, $w\in B$, and $xw\in E\left(\mathsf{\Gamma }\right)$ such that $d_{\mathsf{\Gamma }}\left(x\right)=1$, $d_{\mathsf{\Gamma }}\left(u\right)=2$ or 3, $d_{\mathsf{\Gamma }}\left(w\right)=2$ or 3, then $H_f\left({\mathsf{\Gamma }}_1\right)\ge H_f\left(\mathsf{\Gamma }\right)$, where ${\mathsf{\Gamma }}_1=\mathsf{\Gamma }-wx+ux$.

(2) 

If $u\in A$, $w\in B$, and $xw,yw\in E\left(\mathsf{\Gamma }\right)$ such that $d_{\mathsf{\Gamma }}\left(x\right)=d_{\mathsf{\Gamma }}\left(y\right)=1$, $d_{\mathsf{\Gamma }}\left(u\right)=2$, $d_{\mathsf{\Gamma }}\left(w\right)=4$, then $H_f\left({\mathsf{\Gamma }}_2\right)=H_f\left(\mathsf{\Gamma }\right)$, where ${\mathsf{\Gamma }}_2=\mathsf{\Gamma }-wx-wy+ux+uy$.

(3) 

If $u\in A$, $w\in B$, and $xw\in E\left(\mathsf{\Gamma }\right)$ such that $d_{\mathsf{\Gamma }}\left(x\right)=1$, $d_{\mathsf{\Gamma }}\left(u\right)=3$, $d_{\mathsf{\Gamma }}\left(w\right)=4$, then $H_f\left({\mathsf{\Gamma }}_3\right)=H_f\left(\mathsf{\Gamma }\right)$, where ${\mathsf{\Gamma }}_3=\mathsf{\Gamma }-wx+ux$.

(4) 

If $u,v\in B$, and $xu\in E\left(\mathsf{\Gamma }\right)$ such that $d_{\mathsf{\Gamma }}\left(x\right)=1$, $d_{\mathsf{\Gamma }}\left(u\right)=2$ or 3, $d_{\mathsf{\Gamma }}\left(v\right)=3$, then $H_f\left({\mathsf{\Gamma }}_4\right)\ge H_f\left(\mathsf{\Gamma }\right)$, where ${\mathsf{\Gamma }}_4=\mathsf{\Gamma }-ux+vx$.

(5) 

If $u,v,w\in B$, and $xu,yv\in E\left(\mathsf{\Gamma }\right)$ such that $d_{\mathsf{\Gamma }}\left(x\right)=d_{\mathsf{\Gamma }}\left(y\right)=1$, $d_{\mathsf{\Gamma }}\left(u\right)=d_{\mathsf{\Gamma }}\left(v\right)=d_{\mathsf{\Gamma }}\left(w\right)=2$, then $H_f\left({\mathsf{\Gamma }}_5\right)\ge H_f\left(\mathsf{\Gamma }\right)$, where ${\mathsf{\Gamma }}_5=\mathsf{\Gamma }-xu-yv+wx+wy$.

(6) 

If $u,v\in B$, and $xu\in E\left(\mathsf{\Gamma }\right)$ such that $d_{\mathsf{\Gamma }}\left(x\right)=1$, $d_{\mathsf{\Gamma }}\left(u\right)=2$ and $d_{\mathsf{\Gamma }}\left(v\right)=2$, then $H_f\left({\mathsf{\Gamma }}_6\right)\ge H_f\left(\mathsf{\Gamma }\right)$, where ${\mathsf{\Gamma }}_6=\mathsf{\Gamma }-ux+vx$.

Proof. 

Suppose that $\mathsf{\Gamma }$ is the graph of order n and $f\left(x\right)$ is convex.

• For $\left(1\right)$, from Lemma 1, $f(d_{\mathsf{\Gamma }}\left(u\right)+1)-f\left(d_{\mathsf{\Gamma }}\left(u\right)\right)+f(d_{\mathsf{\Gamma }}\left(w\right)-1)-f\left(d_{\mathsf{\Gamma }}\left(w\right)\right)\ge 0$ holds for $d_{\mathsf{\Gamma }}\left(u\right)=2,3$ and $d_{\mathsf{\Gamma }}\left(w\right)=2,3$, and hence

$$H_f\left({\mathsf{\Gamma }}_1\right)=H_f\left(\mathsf{\Gamma }\right)+f(d_{\mathsf{\Gamma }}\left(u\right)+1)-f\left(d_{\mathsf{\Gamma }}\left(u\right)\right)+f(d_{\mathsf{\Gamma }}\left(w\right)-1)-f\left(d_{\mathsf{\Gamma }}\left(w\right)\right)\ge H_f\left(\mathsf{\Gamma }\right).$$

For $\left(2\right)$, we can easily obtain

$$\begin{array}{cc}\hfill H_f\left({\mathsf{\Gamma }}_2\right)& =H_f\left(\mathsf{\Gamma }\right)+f(d_{\mathsf{\Gamma }}\left(u\right)+2)-f\left(d_{\mathsf{\Gamma }}\left(u\right)\right)+f(d_{\mathsf{\Gamma }}\left(w\right)-2)-f\left(d_{\mathsf{\Gamma }}\left(w\right)\right)\hfill \\ \hfill & =H_f\left(\mathsf{\Gamma }\right)+f\left(4\right)-f\left(2\right)+f\left(2\right)-f\left(4\right)=H_f\left(\mathsf{\Gamma }\right).\hfill \end{array}$$

For $\left(3\right)$, we have

$$\begin{array}{cc}\hfill H_f\left({\mathsf{\Gamma }}_3\right)& =H_f\left(\mathsf{\Gamma }\right)+f(d_{\mathsf{\Gamma }}\left(u\right)+1)-f\left(d_{\mathsf{\Gamma }}\left(u\right)\right)+f(d_{\mathsf{\Gamma }}\left(w\right)-1)-f\left(d_{\mathsf{\Gamma }}\left(w\right)\right)\hfill \\ \hfill & =H_f\left(\mathsf{\Gamma }\right)+f\left(4\right)-f\left(3\right)+f\left(3\right)-f\left(4\right)=H_f\left(\mathsf{\Gamma }\right).\hfill \end{array}$$

For $\left(4\right)$, from Lemma 1, we know that $f(d_{\mathsf{\Gamma }}\left(v\right)+1)-f\left(d_{\mathsf{\Gamma }}\left(v\right)\right)+f(d_{\mathsf{\Gamma }}\left(u\right)-1)-f\left(d_{\mathsf{\Gamma }}\left(u\right)\right)\ge 0$ holds for $d_{\mathsf{\Gamma }}\left(u\right)=2,3$ and $d_{\mathsf{\Gamma }}\left(v\right)=3$, and hence

$$H_f\left({\mathsf{\Gamma }}_4\right)=H_f\left(\mathsf{\Gamma }\right)+f(d_{\mathsf{\Gamma }}\left(v\right)+1)-f\left(d_{\mathsf{\Gamma }}\left(v\right)\right)+f(d_{\mathsf{\Gamma }}\left(u\right)-1)-f\left(d_{\mathsf{\Gamma }}\left(u\right)\right)\ge H_f\left(\mathsf{\Gamma }\right).$$

For $\left(5\right)$, since $f\left(x\right)$ is a convex function, it follows that $f\left(1\right)+f\left(3\right)\ge 2f\left(2\right)$. From Lemma 1, we have $f\left(1\right)+f\left(4\right)\ge f\left(2\right)+f\left(3\right)$, and hence $2f\left(1\right)+f\left(4\right)\ge 3f\left(2\right)$. Then,

$$\begin{array}{cc}\hfill H_f\left({\mathsf{\Gamma }}_5\right)=& H_f\left(\mathsf{\Gamma }\right)+f(d_{\mathsf{\Gamma }}\left(v\right)-1)-f\left(d_{\mathsf{\Gamma }}\left(v\right)\right)+f(d_{\mathsf{\Gamma }}\left(u\right)-1)-f\left(d_{\mathsf{\Gamma }}\left(u\right)\right)\hfill \\ \hfill & +f(d_{\mathsf{\Gamma }}\left(w\right)+2)-f\left(d_{\mathsf{\Gamma }}\left(w\right)\right)\hfill \\ \hfill =& H_f\left(\mathsf{\Gamma }\right)+2f\left(1\right)-2f\left(2\right)+f\left(4\right)-f\left(2\right)\hfill \\ \hfill =& H_f\left(\mathsf{\Gamma }\right)+2f\left(1\right)+f\left(4\right)-3f\left(2\right)\ge H_f\left(\mathsf{\Gamma }\right).\hfill \end{array}$$

For $\left(6\right)$, from Lemma 1, we know that $f(d_{\mathsf{\Gamma }}\left(v\right)+1)-f\left(d_{\mathsf{\Gamma }}\left(v\right)\right)+f(d_{\mathsf{\Gamma }}\left(u\right)-1)-f\left(d_{\mathsf{\Gamma }}\left(u\right)\right)\ge 0$ holds for $d_{\mathsf{\Gamma }}\left(u\right)=2$ and $d_{\mathsf{\Gamma }}\left(v\right)=2$, and hence

$$H_f\left({\mathsf{\Gamma }}_6\right)=H_f\left(\mathsf{\Gamma }\right)+f(d_{\mathsf{\Gamma }}\left(v\right)+1)-f\left(d_{\mathsf{\Gamma }}\left(v\right)\right)+f(d_{\mathsf{\Gamma }}\left(u\right)-1)-f\left(d_{\mathsf{\Gamma }}\left(u\right)\right)\ge H_f\left(\mathsf{\Gamma }\right).$$

□

For chemical trees, we have the following upper bound.

Theorem 7.

Let T be a chemical tree of order $n(n\ge 5)$. If $f\left(x\right)$ is a convex function, then three integers ($m_1,m_2,m_3$) exist such that

$$H_f\left(T\right)\le \left\{\begin{array}{cc}{m}_{1}f\left(4\right)+(n-m_1-1)f\left(1\right)+f\left(2\right)\hfill & \mathrm{if}i=0,\hfill \\ m_{2}f\left(4\right)+(n-m_2-1)f\left(1\right)+f\left(3\right)\hfill & \mathrm{if}i=1,\hfill \\ m_{3}f\left(4\right)+(n-m_3)f\left(1\right)\hfill & \mathrm{if}i=2\hfill \end{array}\right.$$

with equality if and only if T contains only one 2-degree vertex but contains no 3-degree vertices for $i=0$; T contains only one 3-degree vertex but contains no 2-degree vertices for $i=1$; and T only contains 1-degree vertices and 4-degree vertices for $i=2$, where $n\equiv i (\mathrm{mod} 3)$.

Proof. 

Suppose that T is a chemical tree of order n and $f\left(x\right)$ is a convex function. By operations $\left(1\right),\left(2\right)$, and $\left(3\right)$ of Lemma 2, we can obtain a new tree $T^{\prime }$ with $V\left(T^{\prime }\right)=V\left(T\right)$ containing no 2-degree vertices or 3-degree vertices in A. That is to say, all of the 2-degree vertices and 3-degree vertices are in B. Suppose that $n\equiv i (\mathrm{mod} 3)$ and $n_1,n_2,n_3,n_4$ are the number of vertices with degree $1,2,3,4$, respectively, in $T^{\prime }$.

Note that $H_f\left(T^{\prime }\right)\ge H_f\left(T\right)$. We distinguish the following cases to show this theorem.

Case 1.

i = 0.

We claim that $n_2\ne 0$ or $n_3\ne 0$; otherwise, $T^{\prime }$ contains only 1-degree and 4-degree vertices. Since $n_1+n_4=n$ and $n_1+4n_4=2(n-1)$, we have $n=3n_4+2$, contradicting the fact that $n\equiv 0 (\mathrm{mod} 3)$.

Since $n_1+n_2+n_3+n_4=n$ and $n_1+2n_2+3n_3+4n_4=2(n-1)$, we have $n_2+2n_3\equiv 1 (\mathrm{mod} 3)$, and so $n_2-n_3\equiv 1 (\mathrm{mod} 3)$ and $n_3-n_2\equiv 2 (\mathrm{mod} 3)$.

If $n_2\ge n_3$, then it follows from $\left(4\right)$ of Lemma 2 that

$$\begin{array}{cc}\hfill H_f\left(T^{\prime }\right)=& n_{1}f\left(1\right)+n_{2}f\left(2\right)+n_{3}f\left(3\right)+n_{4}f\left(4\right)\hfill \\ \hfill \le & (n_1+n_3)f\left(1\right)+(n_2-n_3)f\left(2\right)+(n_4+n_3)f\left(4\right).\hfill \end{array}$$

Suppose that $n_2-n_3=3k_1+1$. From $\left(5\right)$ of Lemma 2, we have

$$H_f\left(T^{\prime }\right)\le (n_1+n_3+2k_1)f\left(1\right)+f\left(2\right)+(n_4+n_3+k_1)f\left(4\right).$$

Let $m_1=n_4+n_3+k_1$, and thus we are done.

If $n_2<n_3$, then it follows from $\left(4\right)$ of Lemma 2 that $H_f\left(T^{\prime }\right)=n_{1}f\left(1\right)+n_{2}f\left(2\right)+n_{3}f\left(3\right)+n_{4}f\left(4\right)\le$ $(n_1+n_2)f\left(1\right)+(n_3-n_2)f\left(3\right)+(n_4+n_2)f\left(4\right).$ Suppose that $n_3-n_2=3{\ell }_1+2$. From $\left(4\right)$ of Lemma 2, we have

$$\begin{array}{cc}\hfill H_f\left(T^{\prime }\right)& \le (n_1+n_2)f\left(1\right)+({\ell }_1+1)f\left(2\right)+{\ell }_{1}f\left(3\right)+(n_4+n_2+{\ell }_1+1)f\left(4\right)\hfill \\ \hfill & \le (n_1+n_2+{\ell }_1)f\left(1\right)+f\left(2\right)+(n_4+n_2+2{\ell }_1+1)f\left(4\right).\hfill \end{array}$$

Let $m_1=n_4+n_2+2{\ell }_1+1$, and thus we are done.

Case 2.

i = 1.

We claim that $n_2\ne 0$ or $n_3\ne 0$; otherwise, $T^{\prime }$ contains only 1-degree and 4-degree vertices. Since $n_1+n_4=n$ and $n_1+4n_4=2(n-1)$, we have $n=3n_4+2$, contradicting the fact that $n\equiv 1 (\mathrm{mod} 3)$.

Since $n_1+n_2+n_3+n_4=n$ and $n_1+2n_2+3n_3+4n_4=2(n-1)$, we have $n_2+2n_3\equiv 2 (\mathrm{mod} 3)$, and so $n_2-n_3\equiv 2 (\mathrm{mod} 3)$, $n_3-n_2\equiv 1 (\mathrm{mod} 3)$.

If $n_2\ge n_3$, then it follows from $\left(4\right)$ of Lemma 2 that

$$\begin{array}{cc}\hfill H_f\left(T^{\prime }\right)& =n_{1}f\left(1\right)+n_{2}f\left(2\right)+n_{3}f\left(3\right)+n_{4}f\left(4\right)\hfill \\ \hfill & \le (n_1+n_3)f\left(1\right)+(n_2-n_3)f\left(2\right)+(n_4+n_3)f\left(4\right).\hfill \end{array}$$

If $n_2-n_3=3k_2+2$, it follows from $\left(5\right)$ and $\left(6\right)$ of Lemma 2 that

$$\begin{array}{cc}\hfill H_f\left(T^{\prime }\right)& \le (n_1+n_3+2k_2)f\left(1\right)+2f\left(2\right)+(n_4+n_3+k_2)f\left(4\right)\hfill \\ \hfill & \le (n_1+n_3+2k_2+1)f\left(1\right)+f\left(3\right)+(n_4+n_3+k_2)f\left(4\right).\hfill \end{array}$$

Let $m_2=n_4+n_3+k_2$, and thus we are done.

If $n_2<n_3$, then it follows from $\left(4\right)$ of Lemma 2 that

$$\begin{array}{cc}\hfill H_f\left(T^{\prime }\right)& =n_{1}f\left(1\right)+n_{2}f\left(2\right)+n_{3}f\left(3\right)+n_{4}f\left(4\right)\hfill \\ \hfill & \le (n_1+n_2)f\left(1\right)+(n_3-n_2)f\left(3\right)+(n_4+n_2)f\left(4\right).\hfill \end{array}$$

If $n_3-n_2=3{\ell }_2+1$, then it follows from $\left(4\right)$ of Lemma 2 that

$$\begin{array}{cc}\hfill H_f\left(T^{\prime }\right)& \le (n_1+n_2)f\left(1\right)+(3{\ell }_2+1)f\left(3\right)+(n_4+n_2)f\left(4\right)\hfill \\ \hfill & \le (n_1+n_2)f\left(1\right)+{\ell }_{2}f\left(2\right)+({\ell }_2+1)f\left(3\right)+(n_4+n_2+{\ell }_2)f\left(4\right)\hfill \\ \hfill & \le (n_1+n_2+{\ell }_2)f\left(1\right)+f\left(3\right)+(n_4+n_2+2{\ell }_2)f\left(4\right).\hfill \end{array}$$

Let $m_2=n_4+n_2+2{\ell }_2$, and thus we are done.

Case 3.

i = 2.

Since $n_1+n_2+n_3+n_4=n$ and $n_1+2n_2+3n_3+4n_4=2(n-1)$, we have $n_2+2n_3\equiv 0 (\mathrm{mod} 3)$, and so $n_2-n_3\equiv 0 (\mathrm{mod} 3)$, $n_3-n_2\equiv 0 (\mathrm{mod} 3)$.

If $n_2\ge n_3$, then it follows from $\left(4\right)$ of Lemma 2 that $H_f\left(T^{\prime }\right)=n_{1}f\left(1\right)+n_{2}f\left(2\right)+n_{3}f\left(3\right)+n_{4}f\left(4\right)$ $\le (n_1+n_3)f\left(1\right)+(n_2-n_3)f\left(2\right)+(n_4+n_3)f\left(4\right).$ Suppose that $n_2-n_3=3k_3$. By $\left(5\right)$ of Lemma 2, we have

$$H_f\left(T^{\prime }\right)\le (n_1+n_3+2k_3)f\left(1\right)+(n_4+n_3+k_3)f\left(4\right).$$

Let $m_3=n_4+n_3+k_3$, and thus we are done.

If $n_2<n_3$, then it follows from $\left(4\right)$ of Lemma 2 that

$$\begin{array}{cc}\hfill H_f\left(T^{\prime }\right)=& n_{1}f\left(1\right)+n_{2}f\left(2\right)+n_{3}f\left(3\right)+n_{4}f\left(4\right)\hfill \\ \hfill \le & (n_1+n_2)f\left(1\right)+(n_3-n_2)f\left(3\right)+(n_4+n_2)f\left(4\right).\hfill \end{array}$$

Suppose that $n_3-n_2=3{\ell }_3$. By $\left(4\right)$ of Lemma 2, we have

$$\begin{array}{cc}\hfill H_f\left(T^{\prime }\right)& \le (n_1+n_2)f\left(1\right)+3{\ell }_{3}f\left(3\right)+(n_4+n_2)f\left(4\right)\hfill \\ \hfill & \le (n_1+n_2)f\left(1\right)+{\ell }_{3}f\left(2\right)+{\ell }_{3}f\left(3\right)+(n_4+n_2+{\ell }_3)f\left(4\right)\hfill \\ \hfill & \le (n_1+n_2+{\ell }_3)f\left(1\right)+(n_4+n_2+2{\ell }_3)f\left(4\right).\hfill \end{array}$$

Let $m_3=(n_4+n_2+2{\ell }_3)$, and thus we are done. □

For trees, we have the following upper bound.

Theorem 8.

Let T be a tree of order $n(n\ge 4)$. If $f\left(x\right)$ is a convex function, then $H_f\left(T\right)\le (n-1)f\left(1\right)+f(n-1)$ with equality if and only if $T\cong S_n$.

Proof. 

If $n=4$, then by the proof of Theorem 4, we obtain $H_f\left(S_4\right)>H_f\left(P_4\right)$. The result holds for $n=4$.

We now suppose that $n\ge 5$. We prove this result by induction on n. Assume that the result holds for $n-1$ and prove it for n. Let $T^{\prime }$ be a tree of order $n-1$ such that $T-v_j=T^{\prime }$, where $d_T\left(v_j\right)=1$, $v_i=N_T\left(v_j\right)$ and $d_{T^{\prime }}\left(v_i\right)=d_T\left(v_i\right)-1=p-1$, (say). Thus, we have $H_f\left(T^{\prime }\right)\le (n-2)f\left(1\right)+f(n-2)$ with equality if and only if $T^{\prime }\cong S_{n-1}$. One can easily see that

$$H_f\left(T\right)=H_f\left(T^{\prime }\right)+f\left(p\right)-f(p-1)+f\left(1\right).$$

Since $f\left(x\right)$ is a convex function, it follows from Lemma 1 that $f(n-1)-f(n-2)\ge f\left(p\right)-f(p-1)$ with equality if and only if $p=n-1$. Therefore, by the induction hypothesis with the above results, we obtain

$$\begin{array}{cc}\hfill H_f\left(T\right)& =H_f\left(T^{\prime }\right)+f\left(p\right)-f(p-1)+f\left(1\right)\hfill \\ \hfill & \le (n-1)f\left(1\right)+f(n-2)+f\left(p\right)-f(p-1)\hfill \\ \hfill & \le (n-1)f\left(1\right)+f(n-1)\hfill \end{array}$$

and the result holds by induction. Moreover, the equality holds if and only if $T^{\prime }\cong S_{n-1}$ and $d_T\left(v_i\right)=p=n-1$, that is, if and only if $T\cong S_n$. □

Remark 1.

If $f\left(x\right)$ is a convex function, then by Theorems 4 and 8, we conclude that the path $P_n$ gives the minimum $H_f\left(T\right)$ and the star gives the maximum $H_f\left(T\right)$ among all trees of order n.

3. Nordhaus–Gaddum-Type Results

In this section, we give the Nordhaus–Gaddum-type results for the vertex degree function index.

Theorem 9.

Let Γ be a graph of order n. If $f\left(x\right)$ is a convex function, then

$$H_f\left(\mathsf{\Gamma }\right)+H_f\left(\overline{\mathsf{\Gamma }}\right)\ge \left\{\begin{array}{cc}2nf\left(\frac{n-1}{2}\right)\hfill & \mathrm{if}nisodd,\hfill \\ n\left[f\left(\frac{n}{2}\right)+f(\frac{n}{2}-1)\right]\hfill & \mathrm{if}niseven.\hfill \end{array}\right.$$

Moreover, the equality holds if and only if Γ is a $\lfloor \frac{n}{2}\rfloor$-regular graph.

Proof. 

We have

$$\begin{array}{cc}\hfill H_f\left(\mathsf{\Gamma }\right)+H_f\left(\overline{\mathsf{\Gamma }}\right)& =\sum _{i=1}^{n}f\left(d_{\mathsf{\Gamma }}\left(v_i\right)\right)+\sum _{i=1}^{n}f(n-1-d_{\mathsf{\Gamma }}\left(v_i\right))\hfill \\ & =\sum _{i=1}^n\left[f\left(d_{\mathsf{\Gamma }}\left(v_i\right)\right)+f(n-1-d_{\mathsf{\Gamma }}\left(v_i\right))\right].\hfill \end{array}$$

(1)

We consider two cases.

Case 1.

n is odd.

$\mathbf{Case}\mathbf{1}.$

First, we assume that $\frac{n-1}{2}\le d_{\mathsf{\Gamma }}\left(v_i\right)\le n-1$. Setting $x=d_{\mathsf{\Gamma }}\left(v_i\right),a=b=d_{\mathsf{\Gamma }}\left(v_i\right)-\frac{n-1}{2}$ in Lemma 1, we obtain

$$f\left(d_{\mathsf{\Gamma }}\left(v_i\right)\right)-f\left(\frac{n-1}{2}\right)\ge f\left(\frac{n-1}{2}\right)-f(n-d_{\mathsf{\Gamma }}\left(v_i\right)-1),$$

that is,

$$f\left(d_{\mathsf{\Gamma }}\left(v_i\right)\right)+f(n-d_{\mathsf{\Gamma }}\left(v_i\right)-1)\ge 2f\left(\frac{n-1}{2}\right)$$

with equality if and only if $d_{\mathsf{\Gamma }}\left(v_i\right)=\frac{n-1}{2}$. From (1), we obtain

$$H_f\left(\mathsf{\Gamma }\right)+H_f\left(\overline{\mathsf{\Gamma }}\right)\ge 2\sum _{i=1}^{n}f\left(\frac{n-1}{2}\right)=2nf\left(\frac{n-1}{2}\right)$$

with equality if and only if $\mathsf{\Gamma }$ is an $\frac{(n-1)}{2}$-regular graph, that is, if and only if $\mathsf{\Gamma }$ is a $\lfloor \frac{n}{2}\rfloor$-regular graph.

Next, we assume that $0\le d_{\mathsf{\Gamma }}\left(v_i\right)\le \frac{n-1}{2}-1$, that is, $\frac{n-1}{2}<d_{\overline{\mathsf{\Gamma }}}\left(v_i\right)\le n-1$. Setting $x=d_{\overline{\mathsf{\Gamma }}}\left(v_i\right),a=b=d_{\overline{\mathsf{\Gamma }}}\left(v_i\right)-\frac{n-1}{2}$ in Lemma 1, we obtain

$$f\left(d_{\overline{\mathsf{\Gamma }}}\left(v_i\right)\right)-f\left(\frac{n-1}{2}\right)\ge f\left(\frac{n-1}{2}\right)-f\left(n-d_{\overline{\mathsf{\Gamma }}}\left(v_i\right)-1\right),$$

that is,

$$f\left(d_{\overline{\mathsf{\Gamma }}}\left(v_i\right)\right)+f\left(n-d_{\overline{\mathsf{\Gamma }}}\left(v_i\right)-1\right)\ge 2f\left(\frac{n-1}{2}\right)$$

with equality if and only if $d_{\overline{\mathsf{\Gamma }}}\left(v_i\right)=\frac{n-1}{2}$. Hence, $H_f\left(\mathsf{\Gamma }\right)+H_f\left(\overline{\mathsf{\Gamma }}\right)=\sum _{i=1}^n\left[f\left(d_{\mathsf{\Gamma }}\left(v_i\right)\right)+f\left(d_{\overline{\mathsf{\Gamma }}}\left(v_i\right)\right)\right]$ $=\sum _{i=1}^n\left[f\left(n-1-d_{\overline{\mathsf{\Gamma }}}\left(v_i\right)\right)+f\left(d_{\overline{\mathsf{\Gamma }}}\left(v_i\right)\right)\right]\ge 2nf\left(\frac{n-1}{2}\right)$ with equality if and only if $\mathsf{\Gamma }$ is an $\frac{(n-1)}{2}$-regular graph, that is, if and only if $\mathsf{\Gamma }$ is a $\lfloor \frac{n}{2}\rfloor$-regular graph.

Case 2.

n is even.

In this case, first we assume that $\frac{n}{2}\le d_{\mathsf{\Gamma }}\left(v_i\right)\le n-1$. Setting $x=d_{\mathsf{\Gamma }}\left(v_i\right),a=d_{\mathsf{\Gamma }}\left(v_i\right)-\frac{n}{2}+1,b=d_{\mathsf{\Gamma }}\left(v_i\right)-\frac{n}{2}$ in Lemma 1, we obtain

$$f\left(d_{\mathsf{\Gamma }}\left(v_i\right)\right)-f\left(\frac{n}{2}-1\right)\ge f\left(\frac{n}{2}\right)-f(n-d_{\mathsf{\Gamma }}\left(v_i\right)-1)$$

with equality if and only if $d_{\mathsf{\Gamma }}\left(v_i\right)=\frac{n}{2}$, and hence

$$H_f\left(\mathsf{\Gamma }\right)+H_f\left(\overline{\mathsf{\Gamma }}\right)\ge \sum _{i=1}^n\left(f\left(\frac{n}{2}\right)+f\left(\frac{n}{2}-1\right)\right)\ge n\left[f\left(\frac{n}{2}\right)+f\left(\frac{n}{2}-1\right)\right]$$

with equality if and only if $\mathsf{\Gamma }$ is an $\frac{n}{2}$-regular graph, that is, if and only if $\mathsf{\Gamma }$ is a $\lfloor \frac{n}{2}\rfloor$-regular graph.

Next, we assume that $0\le d_{\mathsf{\Gamma }}\left(v_i\right)\le \frac{n}{2}-1$, that is, $\frac{n}{2}\le d_{\overline{\mathsf{\Gamma }}}\left(v_j\right)\le n-1$. Setting $x=d_{\overline{\mathsf{\Gamma }}}\left(v_j\right)$, $a=d_{\overline{\mathsf{\Gamma }}}\left(v_j\right)-\frac{n}{2}+1$, $b=d_{\overline{\mathsf{\Gamma }}}\left(v_j\right)-\frac{n}{2}$ in Lemma 1, we obtain

$$f\left(d_{\overline{\mathsf{\Gamma }}}\left(v_j\right)\right)-f\left(\frac{n}{2}-1\right)\ge f\left(\frac{n}{2}\right)-f\left(n-d_{\overline{\mathsf{\Gamma }}}\left(v_j\right)-1\right),$$

that is,

$$f\left(d_{\overline{\mathsf{\Gamma }}}\left(v_j\right)\right)+f\left(n-d_{\overline{\mathsf{\Gamma }}}\left(v_j\right)-1\right)\ge f\left(\frac{n}{2}\right)+f\left(\frac{n}{2}-1\right)$$

with equality if and only if $d_{\overline{\mathsf{\Gamma }}}\left(v_j\right)=\frac{n}{2}$. Hence,

$$\begin{array}{cc}\hfill H_f\left(\mathsf{\Gamma }\right)+H_f\left(\overline{\mathsf{\Gamma }}\right)=\sum _{i=1}^n\left[f\left(d_{\mathsf{\Gamma }}\left(v_i\right)\right)+f\left(d_{\overline{\mathsf{\Gamma }}}\left(v_i\right)\right)\right]& =\sum _{i=1}^n\left[f\left(n-1-d_{\overline{\mathsf{\Gamma }}}\left(v_i\right)\right)+f\left(d_{\overline{\mathsf{\Gamma }}}\left(v_i\right)\right)\right]\hfill \\ \hfill & \ge n\left[f\left(\frac{n}{2}\right)+f\left(\frac{n}{2}-1\right)\right].\hfill \end{array}$$

with equality if and only if $\mathsf{\Gamma }$ is an $\frac{n}{2}$-regular graph, that is, if and only if $\mathsf{\Gamma }$ is a $\lfloor \frac{n}{2}\rfloor$-regular graph. □

Theorem 10.

Let Γ be a graph of order n with maximum degree Δ. If $f\left(x\right)$ is a convex function, then

$$H_f\left(\mathsf{\Gamma }\right)+H_f\left(\overline{\mathsf{\Gamma }}\right)\le n\left[f(\Delta )+f(n-1-\Delta )\right]$$

with equality if and only if Γ is a regular graph or graph Γ has only two type of degrees Δ and $n-1-\Delta$$(\Delta >(n-1)/2)$.

Proof. 

Setting $x=\Delta$, $a=\Delta -d_{\mathsf{\Gamma }}\left(v_i\right)$ and $b=\Delta +d_{\mathsf{\Gamma }}\left(v_i\right)-(n-1)$, by Lemma 1 we obtain

$$f\left(d_{\mathsf{\Gamma }}\left(v_i\right)\right)+f(n-1-d_{\mathsf{\Gamma }}\left(v_i\right))\le f(\Delta )+f(n-1-\Delta )$$

with equality if and only if $d_{\mathsf{\Gamma }}\left(v_i\right)=n-1-\Delta$ or $i=1$. Hence,

$$\begin{array}{cc}\hfill H_f\left(\mathsf{\Gamma }\right)+H_f\left(\overline{\mathsf{\Gamma }}\right)& =\sum _{i=1}^{n}f\left(d_{\mathsf{\Gamma }}\left(v_i\right)\right)+\sum _{i=1}^{n}f(n-1-d_{\mathsf{\Gamma }}\left(v_i\right))\hfill \\ \hfill & \le \sum _{i=1}^n\left[f(\Delta )+f(n-1-\Delta )\right]=n\left[f(\Delta )+f(n-1-\Delta )\right].\hfill \end{array}$$

Moreover, the equality holds if and only if $d_{\mathsf{\Gamma }}\left(v_i\right)=\Delta$ or $d_{\mathsf{\Gamma }}\left(v_i\right)=n-1-\Delta$ for any vertex $v_i\in V\left(\mathsf{\Gamma }\right)$, that is, if and only if $\mathsf{\Gamma }$ is a regular graph or graph $\mathsf{\Gamma }$ has only two type of degrees $\Delta$ and $n-1-\Delta$$(\Delta >(n-1)/2)$. □

Corollary 2.

Let Γ be a graph of order n. If $f\left(x\right)$ is a convex function, then

$$H_f\left(\mathsf{\Gamma }\right)+H_f\left(\overline{\mathsf{\Gamma }}\right)\le n\left[f(n-1)+f\left(0\right)\right]$$

with equality if and only if Γ is a complete graph or Γ is an empty graph.

Proof. 

Setting $x=n-1$, $a=n-1-\Delta$ and $b=\Delta$, by Lemma 1 we obtain

$$f(\Delta )+f(n-1-\Delta )\le f(n-1)+f\left(0\right)$$

with equality if and only if $\Delta =0$ or $\Delta =n-1$. Using this result with Theorem 10, we obtain the result. Moreover, the equality holds if and only if $\mathsf{\Gamma }$ is a complete graph or $\mathsf{\Gamma }$ is an empty graph. □

The following is the well-known Jensen inequality.

Lemma 3

(Jensen Inequality [21]). If $f\left(x\right)$ is convex function, for all $x_1,x_2,\dots ,x_n\in [a,b]$, then

$$\sum _{i=0}^{n}f\left(x_i\right)\ge nf\left(\frac{\sum _{i=0}^n{x}_i}{n}\right),$$

with equality if and only if $x_1=x_2=\cdots =x_n$.

Theorem 11.

Let Γ be a graph of order n and size m. If $f\left(x\right)$ is a convex function, then

$$H_f\left(\mathsf{\Gamma }\right)+H_f\left(\overline{\mathsf{\Gamma }}\right)\ge nf\left(\frac{2m}{n}\right)+nf\left(\frac{n(n-1)-2m}{n}\right)$$

and

$$H_f\left(\mathsf{\Gamma }\right)·H_f\left(\overline{\mathsf{\Gamma }}\right)\ge n^{2}f\left(\frac{2m}{n}\right)·f\left(\frac{n(n-1)-2m}{n}\right),$$

with equality if and only if Γ is a regular graph.

Proof. 

Since $f\left(x\right)$ is a convex function and $\sum _{i=1}^n{d}_{\mathsf{\Gamma }}\left(v_i\right)=2m$, it follows from Lemma 3 that

$$\sum _{i=1}^{n}f\left(d_{\mathsf{\Gamma }}\left(v_i\right)\right)\ge nf\left(\frac{\sum _{i=1}^n{d}_{\mathsf{\Gamma }}\left(v_i\right)}{n}\right)=nf\left(\frac{2m}{n}\right)$$

and

$$\sum _{i=1}^{n}f\left(d_{\overline{\mathsf{\Gamma }}}\left(v_i\right)\right)\ge nf\left(\frac{\sum _{i=1}^n{d}_{\overline{\mathsf{\Gamma }}}\left(v_i\right)}{n}\right)=nf\left(\frac{n(n-1)-2m}{n}\right).$$

Hence,

$$\begin{array}{cc}\hfill H_f\left(\mathsf{\Gamma }\right)+H_f\left(\overline{\mathsf{\Gamma }}\right)& =\sum _{i=1}^{n}f\left(d_{\mathsf{\Gamma }}\left(v_i\right)\right)+\sum _{i=1}^{n}f\left(d_{\overline{\mathsf{\Gamma }}}\left(v_i\right)\right)\ge nf\left(\frac{2m}{n}\right)+nf\left(\frac{n(n-1)-2m}{n}\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill H_f\left(\mathsf{\Gamma }\right)·H_f\left(\overline{\mathsf{\Gamma }}\right)& =\sum _{i=1}^{n}f\left(d_{\mathsf{\Gamma }}\left(v_i\right)\right)·\sum _{i=1}^{n}f\left(d_{\overline{\mathsf{\Gamma }}}\left(v_i\right)\right)\ge n^{2}f\left(\frac{2m}{n}\right)·f\left(\frac{n(n-1)-2m}{n}\right).\hfill \end{array}$$

Moreover, the equality holds if and only if $d_{\mathsf{\Gamma }}\left(v_i\right)=d_{\mathsf{\Gamma }}\left(v_j\right)$ for $1\le i,j\le n$, which means that $\mathsf{\Gamma }$ is a regular graph. □

Theorem 12.

Let Γ be a graph of order n with maximum degree Δ and minimum degree δ. If $f\left(x\right)$ is an increasing function, then

$$\begin{array}{cc}\hfill H_f\left(\mathsf{\Gamma }\right)·H_f\left(\overline{\mathsf{\Gamma }}\right)\le & {(n-1)}^{2}f(\Delta )f(n-1-\delta )+(n-1)f\left(\delta \right)f(n-1-\delta )\hfill \\ \hfill & +(n-1)f(\Delta )f(n-1-\Delta )+f\left(\delta \right)f(n-1-\Delta )\hfill \end{array}$$

with equality if and only if Γ is a regular graph.

Proof. 

Since $\Delta$ is the maximum degree and $\delta$ is the minimum degree, we can assume that $\Delta =d_{\mathsf{\Gamma }}\left(v_1\right)\ge d_{\mathsf{\Gamma }}\left(v_2\right)\ge \cdots \ge d_{\mathsf{\Gamma }}\left(v_n\right)=\delta$. Moreover, we obtain

$$n-1-\delta \ge n-1-d_{\mathsf{\Gamma }}\left(v_i\right)\ge n-1-\Delta for{v}_i\in V\left(\mathsf{\Gamma }\right).$$

Since $f\left(x\right)$ is an increasing function with the above results, we obtain

$$\begin{array}{cc}\hfill H_f\left(\mathsf{\Gamma }\right)·H_f\left(\overline{\mathsf{\Gamma }}\right)=& \sum _{i=1}^{n}f\left(d_{\mathsf{\Gamma }}\left(v_i\right)\right)·\sum _{i=1}^{n}f(n-1-d_{\mathsf{\Gamma }}\left(v_i\right))=\left[f(\Delta )+\sum _{i=2}^{n-2}f\left(d_{\mathsf{\Gamma }}\left(v_i\right)\right)+f\left(\delta \right)\right]\hfill \\ \hfill & \left[f(n-1-\delta )+\sum _{i=2}^{n-2}f(n-1-d_{\mathsf{\Gamma }}\left(v_i\right))+f(n-1-\Delta )\right]\hfill \\ \hfill \le & \left[f\left(\delta \right)+(n-1)f(\Delta )\right]\left[f(n-1-\Delta )+(n-1)f(n-1-\delta )\right]\hfill \\ \hfill =& {(n-1)}^{2}f(\Delta )f(n-1-\delta )+(n-1)f\left(\delta \right)f(n-1-\delta )\hfill \\ \hfill & +(n-1)f(\Delta )f(n-1-\Delta )+f\left(\delta \right)f(n-1-\Delta ).\hfill \end{array}$$

Moreover, the above equality holds if and only if $d_{\mathsf{\Gamma }}\left(v_i\right)=\Delta$ for all $v_i\in V\left(\mathsf{\Gamma }\right)(i=1,2,\dots ,n-1)$, and $d_{\mathsf{\Gamma }}\left(v_i\right)=\delta$ for all $v_i\in V\left(\mathsf{\Gamma }\right)(i=2,3,\dots ,n)$, and hence $\Delta =\delta$, that is, if and only if $\mathsf{\Gamma }$ is a regular graph. □

Corollary 3.

Let Γ be a graph of order n with maximum degree Δ and minimum degree δ. If $f\left(x\right)$ is an increasing function, then

$$H_f\left(\mathsf{\Gamma }\right)·H_f\left(\overline{\mathsf{\Gamma }}\right)\le n^{2}f(\Delta )f(n-1-\delta )$$

with equality if and only if Γ is a regular graph.

Proof. 

Since $f\left(x\right)$ is an increasing function, we have $f\left(\delta \right)\le f(\Delta )$ and $f(n-1-\Delta )\le f(n-1-\delta )$. Using these results in Theorem 12, we obtain the required result. Moreover, the equality holds if and only if $\mathsf{\Gamma }$ is a regular graph. □

Theorem 13.

Let Γ be a graph of order n with maximum degree Δ and minimum degree δ. If $f\left(x\right)$ is an increasing function, then

$$\begin{array}{cc}\hfill H_f\left(\mathsf{\Gamma }\right)·H_f\left(\overline{\mathsf{\Gamma }}\right)\ge & {(n-1)}^{2}f\left(\delta \right)f(n-1-\Delta )+(n-1)f\left(\delta \right)f(n-1-\delta )\hfill \\ \hfill & +(n-1)f(\Delta )f(n-1-\Delta )+f(\Delta )f(n-1-\delta )\hfill \end{array}$$

with equality if and only if Γ is a regular graph.

Proof. 

The proof is similar to the proof of Theorem 12. Since $f\left(x\right)$ is an increasing function, we obtain

$$\begin{array}{cc}\hfill H_f\left(\mathsf{\Gamma }\right)·H_f\left(\overline{\mathsf{\Gamma }}\right)=& \left[f(\Delta )+\sum _{i=2}^{n-2}f\left(d_{\mathsf{\Gamma }}\left(v_i\right)\right)+f\left(\delta \right)\right][f(n-1-\delta )+\sum _{i=2}^{n-2}f(n-1-d_{\mathsf{\Gamma }}\left(v_i\right))\hfill \\ \hfill & +f(n-1-\Delta )]\hfill \\ \hfill \ge & \left[f(\Delta )+(n-1)f\left(\delta \right)\right]\left[f(n-1-\delta )+(n-1)f(n-1-\Delta )\right]\hfill \\ \hfill =& {(n-1)}^{2}f\left(\delta \right)f(n-1-\Delta )+(n-1)f\left(\delta \right)f(n-1-\delta )\hfill \\ \hfill & +(n-1)f(\Delta )f(n-1-\Delta )+f(\Delta )f(n-1-\delta ).\hfill \end{array}$$

Moreover, the equality holds if and only if $\mathsf{\Gamma }$ is a regular graph. □

Corollary 4.

Let Γ be a graph of order n with maximum degree Δ and minimum degree δ. If $f\left(x\right)$ is an increasing function, then

$$H_f\left(\mathsf{\Gamma }\right)·H_f\left(\overline{\mathsf{\Gamma }}\right)\ge n^{2}f\left(\delta \right)f(n-1-\Delta )$$

with equality if and only if Γ is a regular graph.

4. Concluding Remarks

In this report, the vertex-degree function index $H_f\left(\mathsf{\Gamma }\right)$ has been investigated for a different class of graphs. Tight bounds of the vertex-degree function index$H_f\left(\mathsf{\Gamma }\right)$ have been set up for any n vertex-connected graphs, trees, and chemical trees. The extremal graphs where the bounds attain have also been identified. Moreover, we present the Nordhaus–Gaddum-type results for $H_f\left(\mathsf{\Gamma }\right)+H_f\left(\overline{\mathsf{\Gamma }}\right)$ and $H_f\left(\mathsf{\Gamma }\right)·H_f\left(\overline{\mathsf{\Gamma }}\right)$, and the characterization of the extremal graphs. We now pose the following problem related to the work presented in this paper, as a potential topic for further research .

Problem 1.

To find the lower and upper bounds on the vertex-degree function index $H_f\left(\mathsf{\Gamma }\right)$ and characterize corresponding extremal graphs for other significant classes of graphs such as bicyclic, tricyclic graphs, etc.