NEW BOUNDS FOR LAPLACIAN ENERGY

Introduction/purpose: The Laplacian energy (LE) is the sum of absolute values of the terms μi-2m/n, where μi , i=1,2,...,n, are the eigenvalues of the Laplacian matrix of the graph G with n vertices and m edges. The basic results of the theory of LE are outlined, and some new obtained. Methods: Spectral theory of Laplacian matrices is applied. Results: A new class of lower bounds for LE is derived. Conclusion: The paper contributes to the Laplacian spectral theory and tp the theory of graph energies.


Introduction
Throughout this paper, we are concerned with simple graphs, i.e. graphs without directed, multiple, or weighted edges, and without loops. Let G be such a graph, possessing n vertices and m edges. For details of the graph theory see (Harary, 1969), (Cvetković, 1981).
Let the vertices of the graph G be labeled by v 1 , v 2 , …, v n . Let deg(v i ) be the degree (= number of first neighbors) of the vertex v i . Then the Laplacian matrix of G, denoted by L(G), is the square matrix of the order n, whose (i,j)-element is equal to -1 if the vertices v i and v j are adjacent, it is 0 when the vertices v i and v j are not adjacent, and deg(v i ) if i=j. The eigenvalues of L(G), denoted bu μ i , i=1,2,…,n, form the Laplacian spectrum of the graph G. For details of the theory of Laplacian matrices and their spectra see (Grone et al, 1990), (Merris, 1994).
The (ordinary) energy of a graph is defined as the sum of absolute values of the eigenvalues of the adjacency matrix (Li et al, 2012), (Gutman & Furtula, 2019). Extending this concept to Laplacian eigenvalues, the Laplacian energy was defined as (Gutman & Zhou, 2006): (1) For details on the mathematical properties of the Laplacian energy see (Andriantiana, 2016), (Gutman & Furtula, 2019).

Preparations and the main results
The Laplacian eigenvalues μ i , i=1,2,…,n, are non-negative real numbers. If the underlying graph G is connected, then exactly one of these eigenvalues is equal to zero (and the other n-1 eigenvalues are positive-valued). The following relations are well known (Grone et al, 1990). At this point, we note that the sum of squares of vertex degrees is the much studied first Zagreb index; for details see (Borovićanin et al, 2017) and the references cited therein. Combining Eqs. (2) and (3), we directly obtain Numerous upper and lower bounds for the Laplacian energy are known (Gutman & Zhou, 2006), (Andriantiana, 2016), of which we mention here In this paper, we offer four new lower bounds for LE, namely For connected graphs with at least four vertices, all bounds (6)-(9) are strict.
The meaning of the parameters a and b is explained in the subsequent section. Observe that the multiplier in (8) and (9) is the ratio between the geometric and arithmetic means of a and b.

Proofs of bounds (6)-(9)
In order to avoid trivialities, we assume that the graphs considered are connected and have more than three vertices. Let 2 i i m X n    and label the Laplacian eigenvalues of the considered graph so that holds for all i=1,2,…,n, and is strictly negative for at least one value of i. Summing (10) over all i, and bearing in mind Eqs. (1) and (4), we get from which inequality (6) directly follows. For connected graphs, exactly one Laplacian eigenvalue is equal to zero. Therefore, one X-value is equal to 2m/n. Let this be X # .
If we sum (10) over all i, except # i  , then we arrive at which implies inequality (7). Applying the relation 2 P Q PQ   , which holds for any positive real numbers P,Q, with equality if and only if P=Q, we can transfer inequality (6) into inequality (8). In the very same manner, bound (9) is obtained from (7).
It is worth noting that the lower bound (8) has a similar algebraic form as the upper bound (5). Thus (5) and (8) estimate the Laplacian energy from both sides in an analogous, McClelland-type manner.
It can be shown that among the lower bounds (6)-(9), bound (7) is the best. In addition, (6) is better than (8), whereas (7) is better than (9). Numerical testing shows that bound (6) is sharper than {9), However, to verify this by exact mathematical methods seems to be a tough task and remains an open problem.
A special case: regular graphs A graph is said to be regular if all its vertices have equal degrees. Let, thus, the considered graph G be regular, and let deg(v i )=r for all i=1,2,…,n. Then For regular graphs, the Laplacian and ordinary energies coincide. Therefore, bounds (6a)-(9a) hold also for the ordinary energy of regular graphs. Bounds of this kind (for ordinary graph energy) were recently communicated (Oboudi, 2019), (Gutman, 2019).