Bounds on graph energy and Randić energy

Let G be a simple connected graph. Denote by n and m the number of vertices and edges of G, respectively. Let V (G) = {v1, v2, . . . , vn} be the set of the vertices of G and di be the degree of the vertex vi ∈ V (G), i = 1, 2, . . . , n. If vi and vj are two adjacent vertices of G, then it is denoted by i ∼ j. Let ∆ and δ be the maximum and minimum vertex degrees of G, respectively. Let us denote by A = A (G) the adjacency matrix of a graph G. The eigenvalues λ1 ≥ λ2 ≥ · · · ≥ λn of A represent the eigenvalues of G [6]. As well known in spectral graph theory, λ1 is the spectral radius of G and [6]


Introduction
Let G be a simple connected graph. Denote by n and m the number of vertices and edges of G, respectively. Let V (G) = {v 1 , v 2 , . . . , v n } be the set of the vertices of G and d i be the degree of the vertex v i ∈ V (G), i = 1, 2, . . . , n. If v i and v j are two adjacent vertices of G, then it is denoted by i ∼ j. Let ∆ and δ be the maximum and minimum vertex degrees of G, respectively.
Let us denote by A = A (G) the adjacency matrix of a graph G. The eigenvalues λ 1 ≥ λ 2 ≥ · · · ≥ λ n of A represent the eigenvalues of G [6]. As well known in spectral graph theory, λ 1 is the spectral radius of G and [6] A graph G is called as non-singular if no eigenvalue of G is equal to zero. For non-singular graphs, it is obvious that det A = 0. A graph G is singular if at least one of its eigenvalue is equal to zero. Then, det A = 0. The energy of a graph G was defined in [12] as This graph invariant is utilized to estimate the total π−electron energy of a molecule represented by a (molecular) graph. [13,22]. A vast literature exists on E (G), for survey and comprehensive information, see [2,11,14,19,23]. Recently, energy of non-singular graphs has also been studied in the literature. In [8], Das et al. obtained a lower bound on energy of non-singular graphs that improves the lower bounds in [3,22], under certain conditions. Gutman and Das [15] established upper bounds on energy of non-singular (bipartite) molecular graphs. In [15], it was also stated that the upper bound obtained on energy of non-singular molecular graphs improves the upper bound in [3].
The Randić matrix R = R (G) of a graph G is defined so that its (i, j) − th entry is equal to 1/ d i d j if i ∼ j and is equal to 0 otherwise [1]. The eigenvalues ρ 1 ≥ ρ 2 ≥ · · · ≥ ρ n of R are called as the Randić eigenvalues of G [1]. Some well known results concerning the Randić eigenvalues are [1,16] is the general Randić index of the graph G [4,18].
In full analogous manner with the graph energy [12], the Randić energy of G was introduced in [1]. It was defined as [1] For details on the properties and bounds of RE, see the recent works [1,9,10,16,17,20,21,23].
The following upper bound on RE (G) was obtained in [17,21] In the present paper, we find new lower and upper bounds on energy and Randić energy of non-singular (bipartite) graphs. We also show that our lower bounds are stronger than two previously known lower bounds given in [7,9,14,17].

Lemmas
We now list some lemmas that will be needed for our main results.

Lemma 2.3. [10]
Let G be a graph with n vertices and without isolated vertices. Then, for each i = 1, 2, . . . , n where ∆ and δ denote, respectively, the maximum and minimum vertex degrees of G.

Lemma 2.4. [10]
Let G be a graph with n vertices and without isolated vertices and let λ 1 be its spectral radius. Then where ∆ and δ denote, respectively, the maximum and minimum vertex degrees of G.
Lemma 2.6. Let G be a bipartite graph with n vertices and without isolated vertices and let λ 1 be its spectral radius. Then where ∆ and δ denote, respectively, the maximum and minimum vertex degrees of G.
Lemma 2.7. [16] Let G be a graph with n vertices, adjacency matrix A and Randić matrix R. If A has n + , n 0 and n − positive, zero and negative eigenvalues, respectively (n + + n 0 + n − = n), then R has n + , n 0 and n − positive, zero and negative eigenvalues, respectively.
For a graph G with n vertices, the following relation between the determinants of its adjacency and Randić matrices was also given in [16].

Main results
Theorem 3.1. Let G be a connected non-singular graph with n ≥ 2 vertices and m edges. Then Proof. We first recall that |λ i | > 0, 1 ≤ i ≤ n, for a non-singular graph G. Let r = E(G) n and x i = |λi| r − 1, for 1 ≤ i ≤ n. Observe that x i > −1. By means of Equations (1)-(3), we also have This leads to the lower bound (8).
For a non-singular graph G of order n, the following lower bound on E (G) was found in [7,14] E (G) ≥ n (|det A|) 1/n .
Note that f is decreasing for 0 ≤ x < 1 [25]. Thus, f (b) ≤ f (0) = 1, this implies that the lower bound (8) is stronger than the lower bound (10) for connected non-singular graphs. Further, if G is the graph K 2 , then the equality in (8) holds.
Theorem 3.2. Let G be a connected non-singular graph with n ≥ 2 vertices, m edges and maximum vertex degree ∆. Then The equality in (11) is achieved for G ∼ = K n .
Proof. At first, recall that the following inequality for x > 0 [24]. Obviously, |λ i | > 0, 1 ≤ i ≤ n, for a non-singular graph G. Considering these facts with Equation (2), we have Let us consider the function f (x), defined by It is not difficult to see that f is a decreasing function in the interval 1 ≤ x ≤ ∆. Notice that λ 1 ≥ 2m n [6] and 2m n is the average of the vertex degrees that is inevitably greater than unity for connected (molecular) graphs [15]. These together with Lemma 2.2 imply that 1 ≤ 2m n ≤ λ 1 ≤ ∆. Therefore, we have Based on this inequality and Equation (12), we obtain the upper bound in (11). Moreover, one can readily check that the equality in (11) is achieved for G ∼ = K n .
Proof. Notice that x ≤ 1 + x ln x, for x > 0 [24]. Further, |λ i | > 0, 1 ≤ i ≤ n, for non-singular graphs and λ 1 = −λ n , for bipartite graphs [6]. Taking into account these with Equation (2), we obtain Let It can be readily seen that f is a decreasing function in the interval 1 ≤ x ≤ ∆. Recall from Theorem 3.2 that both 2m n and λ 1 belong to this interval and λ 1 ≥ 2m n [6]. Thus, Combining this with Equation (14), we get the required result in (13).
In the next theorem, we give a lower bound on Randić energy of non-singular graphs considering the similar techniques in Theorem 3.1 together with Equations (4)-(6) and Lemmas 2.1, 2.5 and 2.7. Therefore, its proof is omitted.
Remark 3.2. Let c be defined by Equation (16). Observe that 0 ≤ c < 1, since G is connected non-singular graph with n ≥ 3 vertices and the fact that [17,20,21] Consider the function f (x) defined as follows Notice that f is decreasing for 0 ≤ x < 1 [26]. Then f (c) ≤ f (0) = 1. Combining this with Lemma 2.8, we deduce that the lower bound (15) is stronger than the lower bound (17) for connected non-singular graphs. Furthermore, if G is the complete graph K n , then the equality in (15) is attained.
Theorem 3.5. Let G be a connected non-singular graph with n ≥ 2 vertices, m edges, maximum vertex degree ∆ and minimum vertex degree δ. Then The equality in (18) is achieved for G ∼ = K n .
Hence the upper bound in (18) holds. Moreover, it is elementary to check that the equality in (18) is achieved for G ∼ = K n .