On the Sum of the Powers of Distance Signless Laplacian Eigenvalues of Graphs
- PDF / 182,339 Bytes
- 21 Pages / 612 x 792 pts (letter) Page_size
- 94 Downloads / 203 Views
DOI: 10.1007/s13226-020-0455-z
ON THE SUM OF THE POWERS OF DISTANCE SIGNLESS LAPLACIAN EIGENVALUES OF GRAPHS S. Pirzada∗ , Hilal A. Ganie∗ , A. Alhevaz∗∗ and M. Baghipur∗∗∗ ∗ Department ∗∗ Faculty
of Mathematics, University of Kashmir, Srinagar, India
of Mathematical Sciences, Shahrood University of Technology, P.O. Box: 316-3619995161, Shahrood, Iran
∗∗∗ Department
of Mathematics, University of Hormozgon, PO Box 3995, Bandar Abbas, Iran
e-mails: [email protected]; [email protected] [email protected]; [email protected] (Received 24 June 2018; after final revision 27 February 2019; accepted 18 June 2019) Let G be a connected graph with n vertices, m edges and having distance signless Laplacian Pn eigenvalues ρ1 ≥ ρ2 ≥ . . . ≥ ρn ≥ 0. For any real number α 6= 0, let mα (G) = i=1 ρα i be the sum of αth powers of the distance signless Laplacian eigenvalues of the graph G. In this paper, we obtain various bounds for the graph invariant mα (G), which connects it with different parameters associated to the structure of the graph G. We also obtain various bounds for the quantity DEL(G), the distance signless Laplacian-energy-like invariant of the graph G. These bounds improve some previously known bounds. We also pose some extremal problems about DEL(G). Key words : Graph; distance signless Laplacian matrix; distance signless Laplacian eigenvalues; transmission regular. 2010 Mathematics Subject Classification : 05C12, 05C50.
1. I NTRODUCTION Throughout the paper, we consider G as a simple connected graph with vertex set V (G) and edge set E(G). We use standard terminology; for concepts not defined here, we refer the reader to any standard graph theory monograph, such as [15].
1144
S. PIRZADA et al.
Given a simple graph G with n vertices, m edges having vertex set V (G) = {v1 , v2 , . . . , vn } and edge set E(G) = {e1 , e2 , . . . , em }, the adjacency matrix A = (aij ) of G is a (0, 1)-square matrix of order n whose (i, j)-entry is equal to 1, if vi is adjacent to vj ; and equal to 0, otherwise. If Deg(G) = diag(d1 , d2 , . . . , dn ) is the diagonal matrix of the vertex degrees di = dG (vi ), i = 1, 2, . . . , n associated to G, the matrices L(G) = Deg(G) − A(G) and Q(G) = Deg(G) + A(G) are respectively the Laplacian and the signless Laplacian matrices and their spectrum are respectively the Laplacian spectrum and signless Laplacian spectrum of the graph G. These matrices are real symmetric and positive semi-definite. We let 0 = µn ≤ µn−1 ≤ · · · ≤ µ1 and 0 ≤ qn ≤ qn−1 ≤ · · · ≤ q1 to be the Laplacian spectrum and signless Laplacian spectrum of G, respectively. The distance dG (u, v) (or shortly duv ) between the vertices u and v in G is the length of any shortest path in G connecting u and v. When the graph is clear from the context, we will omit the subscript G from the notation. The diameter of G is the maximum distance between any two vertices of G. The transmission T rG (v) of a vertex v is defined to be the sum of the distances from v to P all other vertices in G, i.e.,
Data Loading...
