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 (u, v) 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 (u, v) 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 (u, v) 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 (u, v) 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 \(F\in \{S,S_t,R,Q,T\}\) [5–7]. More results on the harmonic index can been found in [1–3, 10–12].

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. \(\square\)