Mathematical Sciences

, Volume 11, Issue 3, pp 203–209

# The harmonic index of product graphs

• B. N. Onagh
Open Access
Original research

## Abstract

The harmonic index of a graph G is defined as the sum of the weights $$\frac{2}{\hbox{deg} _G(u)+\hbox{deg} _G(v)}$$ of all edges uv of G, where $$\hbox{deg} _G(u)$$ denotes the degree of a vertex u in G. In this paper, we investigate the harmonic index of Cartesian, lexicographic, tensor, strong, corona and edge corona product of two connected graphs.

## Keywords

Harmonic index Product graphs Inverse degree

05C07 05C76

## Introduction

Throughout this paper, all graphs are finite, simple, undirected and connected. For a graph G, V(G) and E(G) denote the set of all vertices and edges, respectively. We will use $$P_n$$, $$C_n$$ and $$K_n$$ to denote the path, the cycle and the complete graph of order n, respectively.

The Cartesian product $$G_1 \Box G_2$$ of graphs $$G_1$$ and $$G_2$$ is the graph with vertex set $$V (G_1) \times V (G_2)$$ in which (uv) is adjacent to $$(u', v')$$ if and only if (1) $$u =u'$$ and $$vv' \in E(G_2)$$, or (2) $$v=v'$$ and $$uu'\in E(G_1)$$.

The lexicographic product (or composition) $$G_1 [ G_2]$$ of graphs $$G_1$$ and $$G_2$$ is the graph with vertex set $$V (G_1) \times V (G_2)$$ in which (uv) is adjacent to $$(u', v')$$ if and only if (1) $$u u'\in E(G_1)$$, or (2) $$u=u'$$ and $$vv'\in E(G_2)$$.

The tensor (or direct) product $$G_1 \times G_2$$ of graphs $$G_1$$ and $$G_2$$ is the graph with vertex set $$V (G_1) \times V (G_2)$$ in which (uv) is adjacent to $$(u', v')$$ if and only if $$u u'\in E(G_1)$$ and $$vv'\in E(G_2)$$.

The strong (or normal) product $$G_1 \boxtimes G_2$$ of graphs $$G_1$$ and $$G_2$$ is the graph with vertex set $$V (G_1) \times V (G_2)$$ in which (uv) is adjacent to $$(u', v')$$ if and only if (1) $$u =u'$$ and $$vv' \in E(G_2)$$, or (2) $$v=v'$$ and $$uu'\in E(G_1)$$, or (3) $$u u'\in E(G_1)$$ and $$vv'\in E(G_2)$$. Obviously, $$G_1 \boxtimes G_2= (G_1 \Box G_2) \cup (G_1 \times G_2)$$.

Let $$V(G_1)=\{ v_1,\ldots , v_{n_1}\}$$. The corona product $$G_1\circ G_2$$ of disjoint graphs $$G_1$$ and $$G_2$$ is obtained by taking $$n_1$$ copies of $$G_2$$ and joining each vertex of the ith copy of $$G_2$$ with the vertex $$v_i \in V (G_1)$$.

Let $$E(G_1)=\{e_1,\ldots , e_{m_1}\}$$. The edge corona product $$G_1\bullet G_2$$ of disjoint graphs $$G_1$$ and $$G_2$$ is obtained by taking $$m_1$$ copies of $$G_2$$ and joining each vertex of the ith copy of $$G_2$$ with two end vertices of the edge $$e_i \in E (G_1)$$.

The following propositions easily follow from the definition and structure of product graphs.

### Proposition 1.1

[8, 9] Let $$G_1$$ and $$G_2$$ be two graphs of orders $$n_1$$ and $$n_2$$, respectively. Then
1. (i)

$$\hbox{deg} _{G_1\Box G_2}(u,v) = \hbox{deg} _{G_1}(u) + \hbox{deg} _{G_2}(v)$$,

2. (ii)

$$\hbox{deg} _{G_1[ G_2]}(u,v) = n_2 \hbox{deg} _{G_1}(u) + \hbox{deg} _{G_2}(v)$$,

3. (iii)

$$\hbox{deg} _{G_1\times G_2}(u,v) = \hbox{deg} _{G_1}(u) \hbox{deg} _{G_2}(v)$$,

4. (iv)

$$\hbox{deg} _{G_1 \boxtimes G_2}(u,v) = \hbox{deg} _{G_1}(u) + \hbox{deg} _{G_2}(v) + \hbox{deg} _{G_1}(u) \hbox{deg} _{G_2}(v)$$.

### Proposition 1.2

[8, 9] Let $$G_1$$ and $$G_2$$ be two disjoint graphs of orders $$n_1$$ and $$n_2$$, respectively. Then
1. (i)

$$\hbox{deg} _{G_1 \circ G_2}(u) = \left\{ \begin{array}{ll} \hbox{deg} _{G_1}(u) +n_2 &{} \quad u\in V(G_1)\\ \hbox{deg} _{G_2}(u) +1 &{} \quad u\in V(G_2), \end{array} \right.$$

2. (ii)

$$\hbox{deg} _{G_1 \bullet G_2}(u) =\left\{ \begin{array}{ll} \left( 1+n_2 \right) \hbox{deg} _{G_1}(u) &{} \quad u\in V(G_1)\\ \hbox{deg} _{G_2}(u) +2 &{} \quad u\in V(G_2). \end{array} \right.$$

The inverse degree and harmonic index of a graph G are two important vertex-degree-based indices related to G, were denoted by r(G) and H(G), respectively, and defined as follows:
$$r(G)= \sum _{u\in V(G)} \frac{1}{\hbox{deg} _G(u)}, \qquad H(G)=\sum _{uv\in E(G)} \frac{2}{\hbox{deg} _G(u) +\hbox{deg} _G(v)}.$$
In recent years, the harmonic index has been extensively studied. Shwetha et al. [9] derived expressions for the harmonic index of the join, corona product, Cartesian product, composition and symmetric difference of graphs. Recently, Onagh investigated the harmonic index of subdivision graph S(G), t-subdivision graph $$S_t(G)$$, vertex-semitotal graph R(G), edge-semitotal graph Q(G), total graph T(G) and F-sum of graphs, where $$F\in \{S,S_t,R,Q,T\}$$ [5, 6, 7]. More results on the harmonic index can been found in [1, 2, 3, 10, 11, 12].

In this paper, we study the harmonic index of Cartesian, lexicographic, tensor, strong, corona and edge corona product of two graphs $$G_1$$ and $$G_2$$ and present some bounds in terms of the harmonic index and inverse degree of $$G_1$$ and $$G_2$$.

## Main results

In this section, we give some bounds for the harmonic index of graphs $$G_1\Box G_2$$, $$G_1[ G_2]$$, $$G_1\times G_2$$, $$G_1 \boxtimes G_2$$, $$G_1\circ G_2$$ and $$G_1\bullet G_2$$ in terms of $$H(G_1)$$, $$H(G_2)$$, $$r(G_1)$$ and $$r(G_2)$$. To do this, we need the following well-known inequality.

Jensen’s inequality [4] Let f be a convex function on the interval I and $$x_1, \dots ,x_n \in I$$. Then
$$f\left( \frac{x_1+\cdots + x_n}{n}\right) \le \frac{f(x_1)+\cdots + f(x_n)}{n},$$
with equality if and only if $$x_1=\cdots =x_n$$.

Hereafter, $$G_1$$ and $$G_2$$ are two nontrivial graphs with $$|V(G_i)|=n_i$$ and $$|E(G_i)|=m_i$$, $$1\le i\le 2$$.

### Theorem 2.1

Let $$G_1$$ and $$G_2$$ be two graphs. Then
$$H(G_1 \Box G_2)\le \frac{1}{4} \left( n_2 H(G_1) + n_1 H(G_2) + m_2 r(G_1) +m_1 r(G_2) \right),$$
with equality if and only if $$G_1$$ and $$G_2$$ are k-regular graphs.

### Proof

By definition of the harmonic index, we have
\begin{aligned} H(G_1 \Box G_2) & =\sum _{u\in V(G_1)} \sum _{vv'\in E(G_2)} \frac{2}{\hbox{deg} _{G_1 \Box G_2}(u,v) + \hbox{deg} _{G_1 \Box G_2}(u,v')} \\ & \quad + \sum _{v\in V(G_2)} \sum _{uu'\in E(G_1)} \frac{2}{\hbox{deg} _{G_1 \Box G_2}(u,v) + \hbox{deg} _{G_1 \Box G_2}(u',v)} \\ & =\sum _{u\in V(G_1)} \sum _{vv'\in E(G_2)} \frac{2}{ \left( \hbox{deg} _{G_1}(u) + \hbox{deg} _{G_2}(v) \right) + \left( \hbox{deg} _{G_1}(u) +\hbox{deg} _{G_2}(v') \right) } \\ &\quad +\sum _{v\in V(G_2)} \sum _{uu'\in E(G_1)} \frac{2}{ \left( \hbox{deg} _{G_1}(u) + \hbox{deg} _{G_2}(v) \right) + \left( \hbox{deg} _{G_1}(u') +\hbox{deg} _{G_2}(v)\right) } \\ & =\sum _{u\in V(G_1)} \sum _{vv'\in E(G_2)} \frac{2}{ 2 \hbox{deg} _{G_1}(u) + \left( \hbox{deg} _{G_2}(v) +\hbox{deg} _{G_2}(v') \right) } \\ &\quad +\sum _{v\in V(G_2)} \sum _{uu'\in E(G_1)} \frac{2}{ \left( \hbox{deg} _{G_1}(u) + \hbox{deg} _{G_1}(u')\right) + 2 \hbox{deg} _{G_2}(v)} \\ & :=\sum 1 + \sum 2.\end{aligned}
By Jensen’s inequality, for every $$u\in V(G_1)$$ and $$vv' \in E(G_2)$$, we have
$$\frac{2}{ 2 \hbox{deg} _{G_1}(u) + \left( \hbox{deg} _{G_2}(v) +\hbox{deg} _{G_2}(v') \right) } \le \frac{1}{4} \frac{1}{\hbox{deg} _{G_1}(u)} +\frac{1}{4} \frac{2}{\hbox{deg} _{G_2}(v) +\hbox{deg} _{G_2}(v')},$$
(1)
with equality if and only if $$2 \hbox{deg} _{G_1}(u) = \hbox{deg} _{G_2}(v) +\hbox{deg} _{G_2}(v')$$.
Similarly, for every $$v\in V(G_2)$$ and $$uu' \in E(G_1)$$,
$$\frac{2}{ \left( \hbox{deg} _{G_1}(u) + \hbox{deg} _{G_1}(u') \right) + 2 \hbox{deg} _{G_2}(v) } \le \frac{1}{4} \frac{2}{\hbox{deg} _{G_1}(u) + \hbox{deg} _{G_1}(u') } + \frac{1}{4} \frac{1}{\hbox{deg} _{G_2}(v)},$$
(2)
with equality if and only if $$\hbox{deg} _{G_1}(u) + \hbox{deg} _{G_1}(u') = 2 \hbox{deg} _{G_2}(v)$$.
Thus,
\begin{aligned} \sum 1 & \le \frac{1}{4} \sum _{u\in V(G_1)} \sum _{vv'\in E(G_2)} \frac{1}{ \hbox{deg} _{G_1}(u) } + \frac{1}{4} \sum _{u\in V(G_1)} \sum _{vv'\in E(G_2)} \frac{2}{\hbox{deg} _{G_2}(v) +\hbox{deg} _{G_2}(v') } \\ & =\frac{1}{4} \sum _{u\in V(G_1)} \left( m_2\times \frac{1}{ \hbox{deg} _{G_1}(u)} \right) + \frac{1}{4} \sum _{u\in V(G_1)} H(G_2) \\ &= \frac{1}{4}m_2 r(G_1)+ \frac{1}{4}n_1H(G_2), \end{aligned}
and
\begin{aligned} \sum 2 &\le \frac{1}{4} \sum _{v\in V(G_2)} \sum _{uu'\in E(G_1)} \frac{2}{\hbox{deg} _{G_1}(u) + \hbox{deg} _{G_1}(u') } + \frac{1}{4} \sum _{v\in V(G_2)} \sum _{uu'\in E(G_1)} \frac{1}{\hbox{deg} _{G_2}(v)} \\ & = \frac{1}{4} \sum _{v\in V(G_2)} H(G_1)+ \frac{1}{4} \sum _{v\in V(G_2)} \left( m_1 \times \frac{1}{\hbox{deg} _{G_2}(v)} \right) \\ & = \frac{1}{4} n_2 H(G_1) + \frac{1}{4} m_1 r(G_2). \end{aligned}
So, $$H(G_1 \Box G_2) \le \frac{1}{4} \left( n_2 H(G_1) + n_1 H(G_2) + m_2 r(G_1) +m_1 r(G_2) \right)$$.

Moreover, equality holds in the above inequality if and only if the inequalities (1) and (2) be equalities, i.e., $$G_1$$ and $$G_2$$ are k-regular. $$\square$$

### Theorem 2.2

Let $$G_1$$ and $$G_2$$ be two graphs. Then
$$H(G_1 [ G_2])< \frac{1}{9} n_2 H(G_1) + \frac{1}{4} n_1 H(G_2) +\frac{1}{4}\frac{m_2}{n_2} r(G_1) +\frac{4}{9}n_2 m_1 r(G_2).$$

### Proof

Note that
\begin{aligned}H(G_1 [G_2] ) &=\sum _{u\in V(G_1)} \sum _{vv'\in E(G_2)} \frac{2}{\hbox{deg} _{G_1 [ G_2]}(u,v) + \hbox{deg} _{G_1 [ G_2]}(u,v')} \\&\quad + \sum _{v\in V(G_2)} \sum _{v'\in V(G_2)} \sum _{uu'\in E(G_1)} \frac{2}{\hbox{deg} _{G_1 [ G_2]}(u,v) + \hbox{deg} _{G_1 [ G_2]}(u',v')} \\& =\sum _{u\in V(G_1)} \sum _{vv'\in E(G_2)} \frac{2}{ \left( n_2\hbox{deg} _{G_1}(u) + \hbox{deg} _{G_2}(v) \right) + \left( n_2\hbox{deg} _{G_1}(u) +\hbox{deg} _{G_2}(v') \right) } \\&\quad +\sum _{v\in V(G_2)} \sum _{v'\in V(G_2)} \sum _{uu'\in E(G_1)} \frac{2}{ \left( n_2 \hbox{deg} _{G_1}(u) + \hbox{deg} _{G_2}(v) \right) + \left( n_2\hbox{deg} _{G_1}(u') +\hbox{deg} _{G_2}(v') \right) } \\& =\sum _{u\in V(G_1)} \sum _{vv'\in E(G_2)} \frac{2}{ 2n_2 \hbox{deg} _{G_1}(u) + \left( \hbox{deg} _{G_2}(v) +\hbox{deg} _{G_2}(v') \right) } \\&\quad +\sum _{v\in V(G_2)} \sum _{v'\in V(G_2)}\sum _{uu'\in E(G_1)} \frac{2}{ n_2\left( \hbox{deg} _{G_1}(u) + \hbox{deg} _{G_1}(u') \right) + \hbox{deg} _{G_2}(v)+ \hbox{deg} _{G_2}(v') } \\& :=\sum 1+\sum 2. \end{aligned}
One can see that for every $$u\in V(G_1)$$ and $$vv' \in E(G_2)$$,
$$\frac{2}{2n_2 \hbox{deg} _{G_1}(u) + \left( \hbox{deg} _{G_2}(v) +\hbox{deg} _{G_2}(v') \right) }\le \frac{1}{4n_2}\frac{1}{\hbox{deg} _{G_1}(u)} +\frac{1}{4}\frac{2}{\hbox{deg} _{G_2}(v) +\hbox{deg} _{G_2}(v')},$$
(3)
with equality if and only if $$2n_2 \hbox{deg} _{G_1}(u) = \hbox{deg} _{G_2}(v) +\hbox{deg} _{G_2}(v')$$.□
Also, for every $$v\in V(G_2)$$, $$v'\in V(G_2)$$ and $$uu'\in E(G_1)$$,
\begin{aligned}&\frac{2}{ n_2 \left( \hbox{deg} _{G_1}(u) + \hbox{deg} _{G_1}(u') \right) + \hbox{deg} _{G_2}(v)+ \hbox{deg} _{G_2}(v')} \nonumber \\&\quad \le \frac{1}{9n_2}\frac{2}{\hbox{deg} _{G_1}(u) + \hbox{deg} _{G_1}(u') } + \frac{2}{9} \frac{1}{\hbox{deg} _{G_2}(v)} + \frac{2}{9} \frac{1}{\hbox{deg} _{G_2}(v')}, \end{aligned}
(4)
with equality if and only if $$n_2 ( \hbox{deg} _{G_1}(u) + \hbox{deg} _{G_1}(u') )=\hbox{deg} _{G_2}(v)= \hbox{deg} _{G_2}(v')$$.
Thus,
$$\sum 1\le \frac{1}{ 4} \frac{m_2}{n_2} r(G_1) + \frac{1}{4} n_1H(G_2), \qquad \sum 2\le \frac{1}{9} n_2 H(G_1) +\frac{4}{9}n_2 m_1 r(G_2).$$
Therefore,
$$H(G_1[G_2])\le \frac{1}{9} n_2 H(G_1) + \frac{1}{4} n_1 H(G_2) +\frac{1}{4} \frac{m_2}{n_2} r(G_1) +\frac{4}{9}n_2 m_1 r(G_2).$$
Now, suppose that equality holds in the above inequality. Then, the inequalities (3) and (4) must be equalities. So, $$G_1$$ and $$G_2$$ are $$k_1$$-regular and $$k_2$$-regular graphs, respectively, such that $$2n_2 k_1 = k_2 + k_2$$ and $$n_2 ( k_1+k_1 )=k_2$$, a contradiction. $$\square$$

### Theorem 2.3

Let $$G_1$$ and $$G_2$$ be two graphs. Then
$$H(G_1 \times G_2)\ge 2 H(G_1) H(G_2),$$
with equality if and only if either $$G_1$$ or $$G_2$$ is a regular graph.

### Proof

By definition of the harmonic index, we have
$$H(G_1 \times G_2) = 2\sum _{uu'\in E(G_1)} \sum _{vv'\in E(G_2)} \frac{2}{ \hbox{deg} _{G_1}(u) \hbox{deg} _{G_2}(v) + \hbox{deg} _{G_1}(u') \hbox{deg} _{G_2}(v') }.$$
Note that for every $$uu'\in E(G_1)$$ and $$vv'\in E(G_2)$$,
\begin{aligned}&\frac{2}{ \left( \hbox{deg} _{G_1}(u) + \hbox{deg} _{G_1}(u') \right) \left( \hbox{deg} _{G_2}(v)+ \hbox{deg} _{G_2}(v') \right) } \\&\quad =\frac{2}{\left( \hbox{deg} _{G_1}(u) \hbox{deg} _{G_2}(v) + \hbox{deg} _{G_1}(u') \hbox{deg} _{G_2}(v') \right) + \left( \hbox{deg} _{G_1}(u) \hbox{deg} _{G_2}(v') + \hbox{deg} _{G_1}(u') \hbox{deg} _{G_2}(v) \right) } \\&\quad \le \frac{1}{4} \left( \frac{2}{\hbox{deg} _{G_1}(u) \hbox{deg} _{G_2}(v) + \hbox{deg} _{G_1}(u') \hbox{deg} _{G_2}(v') } + \frac{2}{\hbox{deg} _{G_1}(u) \hbox{deg} _{G_2}(v') + \hbox{deg} _{G_1}(u') \hbox{deg} _{G_2}(v) } \right) , \end{aligned}
with equality if and only if $$\hbox{deg} _{G_1}(u) \hbox{deg} _{G_2}(v) + \hbox{deg} _{G_1}(u') \hbox{deg} _{G_2}(v') = \hbox{deg} _{G_1}(u) \hbox{deg} _{G_2}(v') + \hbox{deg} _{G_1}(u') \hbox{deg} _{G_2}(v)$$, or, equivalently, $$(\hbox{deg} _{G_1}(u)- \hbox{deg} _{G_1}(u') ) (\hbox{deg} _{G_2}(v) - \hbox{deg} _{G_2}(v') )=0$$. On the other hand,
$$\sum _{uu'\in E(G_1)} \sum _{vv'\in E(G_2)} \frac{2}{ \left( \hbox{deg} _{G_1}(u) + \hbox{deg} _{G_1}(u') \right) \left( \hbox{deg} _{G_2}(v)+ \hbox{deg} _{G_2}(v') \right) }= \frac{1}{2} H(G_1) H(G_2),$$
and
\begin{aligned}&\sum _{uu'\in E(G_1)} \sum _{vv'\in E(G_2)} \frac{2}{\hbox{deg} _{G_1}(u) \hbox{deg} _{G_2}(v) + \hbox{deg} _{G_1}(u') \hbox{deg} _{G_2}(v') } \\&\qquad + \sum _{uu'\in E(G_1)} \sum _{vv'\in E(G_2)} \frac{2}{\hbox{deg} _{G_1}(u) \hbox{deg} _{G_2}(v') + \hbox{deg} _{G_1}(u') \hbox{deg} _{G_2}(v) } \\&\quad = \frac{1}{2}H(G_1 \times G_2)+ \frac{1}{2}H(G_1 \times G_2) \\&\quad =H(G_1 \times G_2). \end{aligned}
This implies that $$H(G_1 \times G_2) \ge 2 H(G_1) H(G_2)$$.□

Moreover, equality holds in the above inequality if and only if for every $$uu'\in E(G_1)$$ and $$vv'\in E(G_2)$$, $$(\hbox{deg} _{G_1}(u)- \hbox{deg} _{G_1}(u')) (\hbox{deg} _{G_2}(v) - \hbox{deg} _{G_2}(v'))=0$$, i.e., either $$G_1$$ or $$G_2$$ is regular. $$\square$$

The following corollary is an immediate consequence of Theorem 2.3.

### Corollary 2.4

1. (i)

For any $$n\ge 3$$ and $$m\ge 3$$$$H(P_n \times C_m)= \frac{4}{3}m + \frac{n-3}{2}m$$,

2. (ii)

for any $$n\ge 3$$ and $$m\ge 2$$$$H(P_n \times K_m)= \frac{4}{3}m + \frac{n-3}{2}m$$,

3. (iii)

for any $$n\ge 3$$ and $$m\ge 3$$$$H(C_n \times C_m)= \frac{nm}{2}$$,

4. (iv)

for any $$n\ge 3$$ and $$m\ge 2$$$$H(C_n \times K_m)=\frac{nm}{2}$$,

5. (v)

for any $$n\ge 2$$ and $$m\ge 2$$$$H(K_n \times K_m)= \frac{nm}{2}$$.

### Theorem 2.5

Let $$G_1$$ and $$G_2$$ be two graphs. Then
\begin{aligned} H(G_1 \boxtimes G_2)&\le \frac{1}{9} \left( \left( n_2 + 2 m_2 + r(G_2) \right) H(G_1)+ \left( n_1 + 2m_1 + r(G_1) \right) H(G_2) \right. \\&\quad\left. + H(G_1\times G_2) + m_2 r(G_1) + m_1 r(G_2) \right) , \end{aligned}
with equality if and only if $$G_1$$ and $$G_2$$ are 1-regular graphs.

### Proof

By definition of the harmonic index, we have
\begin{aligned} H(G_1 \boxtimes G_2)&= \sum _{u\in V(G_1)} \sum _{vv'\in E(G_2)} \frac{2}{\hbox{deg} _{G_1 \boxtimes G_2}(u,v) + \hbox{deg} _{G_1 \boxtimes G_2}(u,v')} \\&\quad + \sum _{v\in V(G_2)} \sum _{uu'\in E(G_1)} \frac{2}{\hbox{deg} _{G_1 \boxtimes G_2}(u,v) + \hbox{deg} _{G_1 \boxtimes G_2}(u',v)} \\ &\quad + 2 \sum _{uu'\in E(G_1)} \sum _{vv'\in E(G_2)} \frac{2}{\hbox{deg} _{G_1 \boxtimes G_2}(u,v) + \hbox{deg} _{G_1 \boxtimes G_2}(u',v')} \\&:= \sum 1+ \sum 2 + \sum 3. \end{aligned}
Then,
\begin{aligned} \sum 1&= \sum _{u\in V(G_1)} \sum _{vv'\in E(G_2)} \frac{2}{ \left( \hbox{deg} _{G_1}(u) + \hbox{deg} _{G_2}(v) + \hbox{deg} _{G_1}(u) \hbox{deg} _{G_2}(v) \right) + \left( \hbox{deg} _{G_1}(u) +\hbox{deg} _{G_2}(v') + \hbox{deg} _{G_1}(u) \hbox{deg} _{G_2}(v') \right) } \\&= \sum _{u\in V(G_1)} \sum _{vv'\in E(G_2)} \frac{2}{ 2 \hbox{deg} _{G_1}(u) + \hbox{deg} _{G_1}(u) \left( \hbox{deg} _{G_2}(v) + \hbox{deg} _{G_2}(v')\right) + \left( \hbox{deg} _{G_2}(v) + \hbox{deg} _{G_2}(v') \right) }, \end{aligned}
\begin{aligned} \sum 2&= \sum _{v\in V(G_2)} \sum _{uu'\in E(G_1)} \frac{2}{ \left( \hbox{deg} _{G_1}(u) + \hbox{deg} _{G_2}(v) + \hbox{deg} _{G_1}(u) \hbox{deg} _{G_2}(v) \right) + \left( \hbox{deg} _{G_1}(u') +\hbox{deg} _{G_2}(v) + \hbox{deg} _{G_1}(u') \hbox{deg} _{G_2}(v)\right) } \\&= \sum _{v\in V(G_2)} \sum _{uu'\in E(G_1)} \frac{2}{ \left( \hbox{deg} _{G_1}(u) + \hbox{deg} _{G_1}(u') \right) + \hbox{deg} _{G_2}(v) \left( \hbox{deg} _{G_1}(u) + \hbox{deg} _{G_1}(u') \right) + 2\hbox{deg} _{G_2}(v) }, \end{aligned}
\begin{aligned} \sum 3&= 2\sum _{uu'\in E(G_1)} \sum _{vv'\in E(G_2)} \frac{2}{ \left( \hbox{deg} _{G_1}(u) + \hbox{deg} _{G_2}(v) + \hbox{deg} _{G_1}(u) \hbox{deg} _{G_2}(v) \right) + \left( \hbox{deg} _{G_1}(u') +\hbox{deg} _{G_2}(v') + \hbox{deg} _{G_1}(u') \hbox{deg} _{G_2}(v') \right) } \\&= 2\sum _{uu'\in E(G_1)} \sum _{vv'\in E(G_2)} \frac{2}{\left( \hbox{deg} _{G_1}(u) + \hbox{deg} _{G_1}(u') \right) + \left( \hbox{deg} _{G_2}(v)+ \hbox{deg} _{G_2}(v') \right) + \left( \hbox{deg} _{G_1}(u) \hbox{deg} _{G_2}(v) + \hbox{deg} _{G_1}(u') \hbox{deg} _{G_2}(v') \right) }. \end{aligned}
By similar argument as in the proof of Theorem 2.2, one can show that
$$\sum 1 \le \frac{1}{9} m_2 r(G_1) +\frac{1}{9}r(G_1) H(G_2) +\frac{1}{9} n_1 H(G_2),$$
with equality if and only if $$2 \hbox{deg} _{G_1}(u) =\hbox{deg} _{G_1}(u) ( \hbox{deg} _{G_2}(v) + \hbox{deg} _{G_2}(v')) = \hbox{deg} _{G_2}(v) + \hbox{deg} _{G_2}(v')$$, for all $$u\in V(G_1)$$ and $$vv'\in E(G_2)$$,
\begin{aligned} \sum 2 \le \frac{1}{9} n_2 H(G_1) + \frac{1}{9} r(G_2) H(G_1) + \frac{1}{9} m_1 r(G_2) , \end{aligned}
with equality if and only if $$\hbox{deg} _{G_1}(u) + \hbox{deg} _{G_1}(u') =\hbox{deg} _{G_2}(v)( \hbox{deg} _{G_1}(u) + \hbox{deg} _{G_1}(u') ) = 2\hbox{deg} _{G_2}(v)$$, for all $$v\in V(G_2)$$ and $$uu'\in E(G_1)$$, and
\begin{aligned} \sum 3 \le \frac{2}{9} m_2 H(G_1) + \frac{2}{9} m_1 H(G_2) + \frac{1}{9} H(G_1\times G_2) , \end{aligned}
with equality if and only if $$\hbox{deg} _{G_1}(u) + \hbox{deg} _{G_1}(u') = \ \hbox{deg} _{G_2}(v)+ \hbox{deg} _{G_2}(v') = \hbox{deg} _{G_1}(u) \hbox{deg} _{G_2}(v) + \hbox{deg} _{G_1}(u') \hbox{deg} _{G_2}(v')$$, for all $$uu'\in E(G_1)$$ and $$vv'\in E(G_2)$$.□
Therefore,
\begin{aligned} H(G_1 \boxtimes G_2)&\le \frac{1}{9} \left( \left( n_2 + 2 m_2 + r(G_2) \right) H(G_1)+ \left( n_1 + 2m_1 + r(G_1)\right) H(G_2)\right. \\&\quad \left. + H(G_1\times G_2) + m_2 r(G_1) + m_1 r(G_2) \right) . \end{aligned}
It is easy to see that equality holds in the above inequality if and only if $$G_1$$ and $$G_2$$ are 1-regular graphs. $$\square$$

### Theorem 2.6

Let $$G_1$$ and $$G_2$$ be two disjoint graphs. Then
\begin{aligned} H(G_1\circ G_2) < \frac{1}{4}H(G_1) +\frac{1}{4} n_1 H(G_2) +\frac{2}{9}n_2 r(G_1) + \frac{2}{9}n_1 r(G_2) + \frac{1}{4}n_1 m_2 +\frac{1}{4}\frac{m_1}{n_2} + \frac{2}{9} \frac{n_1n_2}{n_2+1}. \end{aligned}

### Proof

Note that
\begin{aligned} H(G_1\circ G_2)&=\sum _{uv\in E(G_1)} \frac{2}{\hbox{deg} _{G_1\circ G_2}(u)+\hbox{deg} _{G_1\circ G_2}(v)} + n_1 \sum _{uv\in E(G_2)} \frac{2}{\hbox{deg} _{G_1\circ G_2}(u)+\hbox{deg} _{G_1\circ G_2}(v)} \\&\quad + \sum _{u\in V(G_1)} \sum _{v\in V(G_2)} \frac{2}{\hbox{deg} _{G_1\circ G_2}(u)+\hbox{deg} _{G_1\circ G_2}(v)} \\&\quad = \sum _{uv\in E(G_1)} \frac{2}{\left( \hbox{deg} _{G_1}(u) + n_2 \right) + \left( \hbox{deg} _{G_1}(v) + n_2 \right) } +n_1 \sum _{uv\in E(G_2)} \frac{2}{\left( \hbox{deg} _{G_2}(u) + 1 \right) + \left( \hbox{deg} _{G_2}(v) + 1 \right) } \\&\quad + \sum _{u\in V(G_1)} \sum _{v\in V(G_2)} \frac{2}{ \left( \hbox{deg} _{G_1}(u) + n_2 \right) + \left( \hbox{deg} _{G_2}(v) + 1 \right) } \\&= \sum _{uv\in E(G_1)} \frac{2}{\left( \hbox{deg} _{G_1}(u) + \hbox{deg} _{G_1}(v) \right) + 2 n_2 } +n_1 \sum _{uv\in E(G_2)} \frac{2}{\left( \hbox{deg} _{G_2}(u) + \hbox{deg} _{G_2}(v) \right) +2} \\&\quad + \sum _{u\in V(G_1)} \sum _{v\in V(G_2)} \frac{2}{ \hbox{deg} _{G_1}(u) + \hbox{deg} _{G_2}(v) + \left( n_2+1 \right) } \\&:= {} \sum 1 +\sum 2+\sum 3. \end{aligned}
By using a similar method, one can verify that
\begin{aligned} \sum 1 \le \frac{1}{4}H(G_1) +\frac{1}{4}\frac{m_1}{n_2}, \end{aligned}
with equality if and only if $$\hbox{deg} _{G_1}(u) + \hbox{deg} _{G_1}(v)= 2 n_2$$, for all $$uv\in E(G_1)$$,
\begin{aligned} \sum 2 \le \frac{1}{4} n_1 H(G_2) + \frac{1}{4} n_1 m_2, \end{aligned}
with equality if and only if $$\hbox{deg} _{G_2}(u) + \hbox{deg} _{G_2}(v)= 2$$, for all $$uv\in E(G_2)$$, and $$\sum 3 < \frac{2}{9}n_2 r(G_1) + \frac{2}{9}n_1 r(G_2) + \frac{2}{9} \frac{n_1n_2}{n_2+1}$$.□
So,
\begin{aligned} H(G_1\circ G_2)< \frac{1}{4}H(G_1) +\frac{1}{4} n_1 H(G_2) +\frac{2}{9}n_2 r(G_1) + \frac{2}{9}n_1 r(G_2) + \frac{1}{4} n_1 m_2 +\frac{1}{4}\frac{m_1}{n_2} + \frac{2}{9} \frac{n_1n_2}{n_2+1}. \end{aligned}
This completes the proof. $$\square$$

### Theorem 2.7

Let $$G_1$$ and $$G_2$$ be two disjoint graphs. Then
\begin{aligned} H(G_1\bullet G_2) < \frac{1}{n_2+1}H(G_1) +\frac{1}{4} m_1 H(G_2) +\frac{8}{9}m_1 r(G_2) + \frac{4}{9} n_2m_1 + \frac{1}{8}m_1 m_2 + \frac{4}{9} \frac{n_1n_2}{n_2+1}. \end{aligned}

### Proof

Note that
\begin{aligned} H(G_1\bullet G_2)& = \sum _{uv\in E(G_1)} \frac{2}{\hbox{deg} _{G_1\bullet G_2}(u)+\hbox{deg} _{G_1\bullet G_2}(v)} +m_1 \sum _{uv\in E(G_2)} \frac{2}{\hbox{deg} _{G_1\bullet G_2}(u)+\hbox{deg} _{G_1\bullet G_2}(v)} \\&\quad + 2 \sum _{uv\in E(G_1)} \sum _{x\in V(G_2)} \left( \frac{2}{ \hbox{deg} _{G_1\bullet G_1}(u)+\hbox{deg} _{G_1\bullet G_2}(x)} + \frac{2}{\hbox{deg} _{G_1\bullet G_1}(v)+\hbox{deg} _{G_1\bullet G_2}(x) } \right) \\ & = \sum _{uv\in E(G_1)} \frac{2}{ \left( 1 + n_2 \right) \hbox{deg} _{G_1}(u) + \left( 1 + n_2 \right) \hbox{deg} _{G_1}(v) } +m_1 \sum _{uv\in E(G_2)} \frac{2}{\left( \hbox{deg} _{G_2}(u) + 2 \right) + \left( \hbox{deg} _{G_2}(v) + 2 \right) } \\&\quad+ 2\sum _{uv\in E(G_1)} \sum _{x\in V(G_2)}\left( \frac{2}{(1+n_2)\hbox{deg} _{G_1}(u)+ \left( \hbox{deg} _{G_2}(x)+2 \right) }+ \frac{2}{(1+n_2)\hbox{deg} _{G_1}(v)+ \left( \hbox{deg} _{G_2}(x)+2 \right) } \right) \\ & = \frac{1}{ 1 + n_2} \sum _{uv\in E(G_1)} \frac{2}{ \hbox{deg} _{G_1}(u) + \hbox{deg} _{G_1}(v) } +m_1 \sum _{uv\in E(G_2)} \frac{2}{\left( \hbox{deg} _{G_2}(u) + \hbox{deg} _{G_2}(v) \right) +4} \\&\quad + 2\sum _{uv\in E(G_1)} \sum _{x\in V(G_2)} \left( \frac{2}{(1+n_2)\hbox{deg} _{G_1}(u)+ \hbox{deg} _{G_2}(x)+2}+ \frac{2}{(1+n_2)\hbox{deg} _{G_1}(v)+ \hbox{deg} _{G_2}(x)+2} \right) \\ & := \frac{1}{ n_2+1} H(G_1) +\sum 1+\sum 2. \end{aligned}
Similarly, one can prove that $$\sum 1 \le \frac{1}{4} m_1 H(G_2) +\frac{1}{8}m_1 m_2$$, with equality if and only if $$\hbox{deg} _{G_1}(u) + \hbox{deg} _{G_1}(v)= 4$$, for all $$uv\in E(G_1)$$.□

Also, $$\sum 2 \le \frac{4}{9} \frac{n_1n_2}{n_2+1}+ \frac{8}{9} m_1r(G_2)+\frac{4}{9}n_2m_1$$, with equality if and only if $$(1+n_2)\hbox{deg} _{G_1}(u)=(1+n_2) \hbox{deg} _{G_1}(v)=\hbox{deg} _{G_2}(x)= 2$$, for all $$uv\in E(G_1)$$ and $$x\in V(G_2)$$.

Therefore,
\begin{aligned} H(G_1\bullet G_2) \le \frac{1}{n_2+1}H(G_1) +\frac{1}{4} m_1 H(G_2) +\frac{8}{9}m_1 r(G_2) +\frac{4}{9}n_2m_1 + \frac{1}{8} m_1m_2 + \frac{4}{9} \frac{n_1n_2}{n_2+1}. \end{aligned}
It is easy to show that equality cannot occur in the above inequality. $$\square$$

## References

1. 1.
Deng, H., Balachandran, S., Ayyaswamy, S.K., Venkatakrishnan, Y.B.: On the harmonic index and the chromatic number of a graph. Discrete Appl. Math. 161, 2740–2744 (2013)
2. 2.
Lv, J.B., Li, J.: On the harmonic index and the matching numbers of trees. Ars Combin. 116, 407–416 (2014)
3. 3.
Lv, J.B., Li, J., Shiu, W.C.: The harmonic index of unicyclic graphs with given matching number. Kragujevac J. Math. 38(1), 173–182 (2014)
4. 4.
Niculescu, C., Persson, L.E.: Convex functions and their applications: a contemporary approach. Springer, New York (2006)
5. 5.
Onagh, B.N.: The harmonic index of subdivision graphs. Trans. Combin. (to appear)Google Scholar
6. 6.
Onagh, B.N.: The harmonic index for $$R$$-sum of graphs (submitted) Google Scholar
7. 7.
Onagh, B.N.: The harmonic index of edge-semitotal graphs, total graphs and related sums. Kragujevac J. Math. (to appear)Google Scholar
8. 8.
Pattabiraman, K., Nagarajan, S., Chendrasekharan, M.: Zagreb indices and coindices of product graphs. J. Prime Res. Math. 10, 80–91 (2015)
9. 9.
Shwetha, B.S., Lokesha, V., Ranjini, P.S.: On the harmonic index of graph operations. Trans. Combin. 4(4), 5–14 (2015)
10. 10.
Zhong, L.: The harmonic index for graphs. Appl. Math. Lett. 25, 561–566 (2012)
11. 11.
Zhong, L.: The harmonic index on unicyclic graphs. Ars Combin. 104, 261–269 (2012)
12. 12.
Zhong, L.: On the harmonic index and the girth for graphs. Roman. J. Inf. Sci. Technol. 16(4), 253–260 (2013)Google Scholar