ON LOWER BOUNDS FOR THE KIRCHHOFF INDEX

Let G be a simple graph of order n ≥ 2 with m edges. Denote by d1 ≥ d2 ≥ · · · ≥ dn > 0 the sequence of vertex degrees and by μ1 ≥ μ2 ≥ · · · ≥ μn−1 > μn = 0 the Laplacian eigenvalues of the graph G. Lower bounds for the Kirchhoff index, Kf(G) = n ∑n−1 i=1 1 μi , are obtained.


Introduction
Let G = (V, E), V = {1, 2, . . ., n}, E = {e 1 , e 2 , . . ., e m } be a simple connected graph of order n ≥ 3 and size m.If vertices i and j are adjacent, we denote it as i ∼ j.Denote by Some well-known properties of the Laplacian eigenvalues are (see for example [3]): where is the first Zagreb index introduced in [11].In the same paper the second Zagreb index, M 2 , and so called forgotten index, F , were defined as More on the invariant F one can find in [7,9].
The Kirchhoff index of a connected graph is defined as (see [14]): where r ij is the effective resistance distance between vertices i and j.The following more appropriate formula from application point of view was put forward in [10] This, in turn, triggered the study of this invariant and its applications in various areas, including spectral graph theory, molecular chemistry, computer science, etc.
Before we proceed, let us define one special class of d-regular graphs Γ d (see [25]).
Let N(i) be a set of all neighborhoods of the vertex i, i.e.
and by ID = ID(G) the graph invariant called inverse degree In this paper we are concerned with the lower bounds of Kf (G) which depend on some of the parameters n, m, ∆, and invariants going further, we recall some results from the literature needed for our subsequent consideration.

Preliminaries
In this section we outline some results for the invariants Kf (G), M 1 , M 2 , F , t and R −1 that will be needed in the remainder of the paper.
In [28] the following result was proved for the Kf (G): Lemma 2.1.[28] Let G be a simple connected graph with n ≥ 2 vertices and m edges.Then with equality if and only if Remark 2.2.We believe that equality in (1) holds also when This only increases importance of the above inequality.
In [23] the following was proved for the general Randić index: Let G be a simple connected graph with n ≥ 3 vertices and m edges.Then, for any real k with the property with equality if and only if k In [13,22,24] for the Forgotten index the following results were established: Lemma 2.4.[13] Let G be a simple graph with n vertices and m edges.Then with equality if and only if G is regular or bidegreed graph.
Lemma 2.5.[24] Let G be a simple connected graph with n ≥ 2 vertices and m edges.Then with equality if and only if G is regular or bidegreed graph.
Lemma 2.6.[22] Let G be a simple connected graph with n ≥ 2 vertices and m edges.Then where and S is a subset of I = {1, 2, . . ., n}which minimizes the expression Equality in (5) holds if and only if L(G) is regular.
In [4] (see also [15]) for the first Zagreb index, M 1 , the following was proved: Lemma 2.7.[4] Let G be a simple connected graph with n ≥ 2 vertices and m edges. Then with equality if and only if G is regular or bidegreed graph.
For the same invariant in [21] the following was proved: Lemma 2.8.[21] Let G be a simple connected graph with n ≥ 2 vertices and m edges.Then where Equality in (7) holds if and only if G is regular.
For the number of spanning trees, t, of a graph the following was proved in [5]: Lemma 2.9.[5] Let G be a simple connected graph with n ≥ 2 vertices and m edges. Then with equality if and only if G ∼ = K n .
For the same invariant in [1] the following was proved: Lemma 2.10.[1] Let G be a simple connected graph with n ≥ 2 vertices and m edges.Then with equality if and only if G ∼ = K n .

Main results
We will first prove one general result for the lower bounds of Kf (G) in terms of one of the invariants R −1 , M 2 , F or M 1 .
Theorem 3.1.Let G be a simple connected graph with n ≥ 2 vertices and m edges. Then Equalities hold if and only if Proof.In [19] the following inequality was proved From the inequality Also, the following holds and Accordingly, we have that From ( 14) and ( 1) inequalities ( 10) -( 13) are obtained.
If in (10) -( 13) invariants R −1 , M 2 , F and M 1 are replaced with corresponding lower bounds, a number of lower bounds for Kf (G) depending on various graph parameters can be obtained.In what follows we will illustrate this.
From ( 10) and ( 2) the following corollary of Theorem 3.1 is obtained.
Corollary 3.2.Let G be a simple connected graph with n ≥ 3 vertices and m edges.
Then for any real k, ρ with equality if and only if k Since according to (15), the following corollary of Theorem 3.1 holds. Then with equality if and only if Corollary 3.8.Let G be a simple connected graph with n ≥ 2 vertices and m edges. Then with equality if and only if Proof.After applying the arithmetic-geometric mean (AG) inequality on (3), i.e. on the inequality is obtained.From this and ( 12) we obtain (18).
From Lemma 2.5 the following corollary of Theorem 3.1 is obtained.
Corollary 3.9.Let G be a simple connected graph with n ≥ 2 vertices and m edges. Then Similarly, from Lemma 5 and ( 12) the following corollary of Theorem 3.1 is obtained.
Corollary 3.10.Let G be a simple connected graph with n ≥ 2 vertices and m edges. Then Proof.The required result is obtained from ( 13) and inequality The following corollary of Theorem 3.1 sets up a lower bound for Kf (G) in terms of parameters n and m and the invariant t.
Corollary 3.15.Let G be a simple connected graph with n ≥ 3 vertices and m edges. Then with equality if and only if G ∼ = K n .
Similarly, the following can be proved: From the above and inequality (10) we obtain the required result.
0 a sequence of vertex degrees, and by ∆ and δ the greatest and the smallest vertex degrees, respectively.Let A be the adjacency matrix of G, and D = diag(d 1 , d 2 , . . ., d n ) the diagonal matrix of its vertex degrees.Then L = D − A is the Laplacian matrix of G. Eigenvalues of L,µ 1 ≥ µ 2 ≥ • • • ≥ µ n−1 > µ n = 0,are the Laplacian eigenvalues of graph G.
and d(i, j) the distance between vertices i and j.Denote by Γ d a set of all d-regular graphs, 1 ≤ d ≤ n − 1, with diameter D = 2 and |N(i) ∩ N(j)| = d.Further, denote by t = t(G) a number of spanning trees of the connected graph