An improved lower bound for the degree Kirchhoff index of bipartite graphs

Abstract For a connected graphGwithn vertices andm edges, the degree Kirchhoff index ofG is defined asKf∗ (G) = 2m ∑n−1 i=1 (γi) , where γ1 ≥ γ2 ≥ · · · ≥ γn−1 > γn = 0 are the normalized Laplacian eigenvalues of G. In this paper, a lower bound on the degree Kirchhoff index of bipartite graphs is established. Also, it is proved that the obtained bound is stronger than a lower bound derived by Zhou and Trinajstić in [J. Math. Chem. 46 (2009) 283–289].


Introduction
Let G = (V (G) , E (G)) be a simple connected graph with n vertices and m edges, where V (G) = {v 1 , v 2 , . . . , v n }. The degree of a vertex v i ∈ V (G) is denoted by d i , where i = 1, 2, . . . , n. If v i and v j are two adjacent vertices of G, then it is written as i ∼ j.
Denote by A (G) and D (G) = diag (d 1 , d 2 , . . . , d n ) the adjacency and the diagonal degree matrix of G, respectively. The Laplacian matrix of G is defined as L (G) = D (G) − A (G) (see [16]). Since G is assumed to be a connected graph, the matrix D (G) −1/2 exists. The normalized Laplacian matrix of G is the matrix defined [8] by The eigenvalues γ 1 ≥ γ 2 ≥ · · · ≥ γ n−1 > γ n = 0 of L (G) represent the normalized Laplacian eigenvalues of G. Details on the spectra of L (G) can be found in [8]. Chen and Zhang [7] introduced the degree Kirchhoff index of a connected graph G as where r ij is the effective resistance distance between the vertices v i and v j of G. In [7], it was also demonstrated that the degree Kirchhoff index can be expressed in terms of normalized Laplacian eigenvalues as follows: Both of the definitions of the graph invariant Kf * (G) given by (1) and (2) are much studied in the chemical and mathematical literature. For survey and details, see [1, 2, 4, 5, 10-12, 14, 15, 17, 18, 20, 21].
In this paper, we present a lower bound on the degree Kirchhoff index of bipartite graphs. In addition, we show that our lower bound improves the lower bound obtained by Zhou and Trinajstić [21].

Lemmas
In this section, we recall a few well-known properties of the normalized Laplacian eigenvalues of graphs. * Corresponding author (srf burcu bozkurt@hotmail.com). Lemma 2.1. [8] Let G be a connected graph with n ≥ 2 vertices. Then, the following properties regarding the normalized Laplacian eigenvalues are valid:

γ 1 ≤ 2 with equality if and only if G is a bipartite graph.
3. For each 1 ≤ i ≤ n, γ i ∈ [0, 2], γ n = 0 and γ n−1 = 0. Lemma 2.2. [9] Let G be a connected graph with n vertices and m edges. Then, where t (G) is the total number of spanning trees of G.

A lower bound for the degree Kirchhoff index of bipartite graphs
We now give an improved lower bound on the degree Kirchhoff index of bipartite graphs.

Equality in (3) holds if and only if α = 1 and G
Proof. For x > 0, the following inequality can be found in the monograph [19] x ≤ 1 + x ln x, where the equality holds if and only if x = 1. For x > 0, the above inequality can be considered as with equality if and only if x = 1. By Lemma 2.1, γ 1 = 2 and γ i > 0, i = 1, 2, . . . , n − 1, since G is a connected bipartite graph. Then, using these results and Lemma 2.2, we have Now, consider the function f (x) = 1 x + ln x. It can be easily seen that this function is increasing in the interval 1 ≤ x ≤ 2. Then for any real α, γ 2 ≥ α ≥ 1, we have that Bearing this fact in mind and using (2) and (4), we obtain that which is the required inequality (3). Now, assume that the equality holds in (3). Then γ 2 = α and γ 3 = · · · = γ n−1 = 1.
Since G is bipartite, by Lemma 2.1, Considering this with the above conditions, we get that γ 2 = α = 1, which implies that G ∼ = K p,q .
Conversely, it is not difficult to show that the equality holds in (3) for the complete bipartite graph K p,q . Hence, the proof is completed.
By Theorem 3.1 and Lemma 2.3, we have the following corollary.
Equality in (5) holds if and only if G ∼ = K p,q (p + q = n).
Remark 3.1. For a connected bipartite graph G with n ≥ 2 vertices and m edges, Zhou and Trinajstić [21] obtained that with equality if and only if G is a complete bipartite graph. Furthermore, for connected bipartite graphs, the following inequality can be obtained from Theorem 3 of [3]: From the above and (5), we conclude that This implies that the lower bound (5) improves the lower bound (6).