On the Signless Laplacian ABC -Spectral Properties of a Graph

Bilal A. Rather, Hilal A. Ganie and Yilun Shang ()
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Bilal A. Rather: Mathematical Sciences Department, College of Science, United Arab Emirates University, Al Ain 15551, United Arab Emirates
Hilal A. Ganie: Department of School Education, Jammu and Kashmir Government, Kashmir 193404, India
Yilun Shang: Department of Computer and Information Sciences, Northumbria University, Newcastle NE1 8ST, UK

Mathematics, 2024, vol. 12, issue 15, 1-23

Abstract: In the paper, we introduce the signless Laplacian A B C -matrix Q ̃ ( G ) = D ¯ ( G ) + A ̃ ( G ) , where D ¯ ( G ) is the diagonal matrix of A B C -degrees and A ̃ ( G ) is the A B C -matrix of G . The eigenvalues of the matrix Q ̃ ( G ) are the signless Laplacian A B C -eigenvalues of G . We give some basic properties of the matrix Q ̃ ( G ) , which includes relating independence number and clique number with signless Laplacian A B C -eigenvalues. For bipartite graphs, we show that the signless Laplacian A B C -spectrum and the Laplacian A B C -spectrum are the same. We characterize the graphs with exactly two distinct signless Laplacian A B C -eigenvalues. Also, we consider the problem of the characterization of the graphs with exactly three distinct signless Laplacian A B C -eigenvalues and solve it for bipartite graphs and, in some cases, for non-bipartite graphs. We also introduce the concept of the trace norm of the matrix Q ̃ ( G ) − t r ( Q ̃ ( G ) ) n I , called the signless Laplacian A B C -energy of G . We obtain some upper and lower bounds for signless Laplacian A B C -energy and characterize the extremal graphs attaining it. Further, for graphs of order at most 6, we compare the signless Laplacian energy and the A B C -energy with the signless Laplacian A B C -energy and found that the latter behaves well, as there is a single pair of graphs with the same signless Laplacian A B C -energy unlike the 26 pairs of graphs with same signless Laplacian energy and eight pairs of graphs with the same A B C -energy.

Keywords: adjacency matrix; Laplacian (signless) matrix; ABC-matrix; Laplacian ABC-matrix; signless Laplacian ABC-matrix (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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