Mathematical Sciences

, Volume 11, Issue 3, pp 203–209 | Cite as

The harmonic index of product graphs

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 

Mathematics Subject Classification

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 graphGare two important vertex-degree-based indices related to G, were denoted byr(G) andH(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 graphS(G), t-subdivision graph\(S_t(G)\), vertex-semitotal graphR(G), edge-semitotal graphQ(G), total graphT(G) andF-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)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Lv, J.B., Li, J.: On the harmonic index and the matching numbers of trees. Ars Combin. 116, 407–416 (2014)MathSciNetMATHGoogle Scholar
  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)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Niculescu, C., Persson, L.E.: Convex functions and their applications: a contemporary approach. Springer, New York (2006)CrossRefMATHGoogle Scholar
  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)MathSciNetGoogle Scholar
  9. 9.
    Shwetha, B.S., Lokesha, V., Ranjini, P.S.: On the harmonic index of graph operations. Trans. Combin. 4(4), 5–14 (2015)MathSciNetGoogle Scholar
  10. 10.
    Zhong, L.: The harmonic index for graphs. Appl. Math. Lett. 25, 561–566 (2012)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Zhong, L.: The harmonic index on unicyclic graphs. Ars Combin. 104, 261–269 (2012)MathSciNetMATHGoogle Scholar
  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

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Authors and Affiliations

  1. 1.Department of MathematicsGolestan UniversityGorganIran

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