1. Introduction
All graphs considered in the following will be simple. Let 
be a graph with vertex set 
and edge set
. The order and size of graph 
are the number of its vertices and number of its edges, respectively. For undefined terminology and notations, we refer the reader to [1].
For a graph
, the harmonic index 
is defined as
It has been found that the harmonic index, which is a special case of general sum-connectivity index, correlates well with the Randić index [2,3] and the 
-electronic energy of benzenoid hydrocarbons [4,5]. In [6], Favaron et al. considered the relation between harmonic index and the eigenvalues of graphs. Zhong [7] found the minimum and maximum values of the harmonic index for connected graphs and trees, and characterized the corresponding extremal graphs. Recently, Wu et al. [8] give a best possible lower bound for the harmonic index of a graph (a triangle-free graph, respectively) with order 
and minimum degree at least two and characterize the extremal graphs. In this work, we will give a best possible lower bound for the harmonic index of a triangle-free graph with order 
and minimum degree at least
. We show the corresponding extremal graph is the complete bipartite graph
.
2. Another Expression for the Harmonic Index
Before we go forwards to investigate the relationship between the Harmonic index and the minimum degree 
of triangle-free graphs, we will give another expression for the Harmonic index in this section, which is vital in sequel.
Let 
be a graphs with order 
and minimum degree
. Denote by 
, the number of edges joining the vertices of degrees 
and
. Denote by 
the number of vertices of degree of
. Then
(1)
(2)
By counting the edges that incident to a vertex of degree
, 
, one obtains
(3)
Substituting Equation (3) back into Equation (2) and performing appropriate rearrangements, we get
(4)
Now, rewriting Equation (1) as
(5)
and combining Equations (4) and (5) so as to eliminate the term
, we arrive at
i.e.,
(6)
(7)
Remark 2.1 From (7), we see that 
for
vertex graph 
and the equality holds if and only if 
is regular.
3. Main Results
First, we give a lower bound for any triangle-free graph with order 
and size
.
Lemma 3.1 For any triangle-free graph 
with order 
and size
, then
where equality holds if and only if 
is of the form 
for some natural numbers
, and
.
Proof. For any edge 
of
, we have 
, or it would contain triangle(s). By (1), we have
equality holds if and only if 
for every
. Thus, if we denote 
for an edge
, then each of the 
neighbors, including
, of 
should has degree
. Similarly, each of the 
neighbors of 
has degree
. Therefore, 
and
.
Lemma 3.2 Let
, for
then 
is decreasing in 
and increasing in
.
Proof. We have
and 
for 
.
Theorem 3.3 Let 
be a triangle-free graph with order 
and the minimum degree 
. Then
where equality holds if and only if
.
Proof. Assume 
is the size of graph
. We divide the proof into the following two cases.
Case 1.
. The result follows by Lemma 3.1.
Case 2.
. Note that the maximum degree 
for graph
, or it would contain triangle(s). By (6) and Lemma 3.2, we have
For equalities to hold above, we must have 
, which means that
.
NOTES