1. On the eigenvalues of Laplacian ABC-matrix of graphs.
- Author
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Rather, Bilal Ahmad, Ganie, Hilal A., and Li, Xueliang
- Subjects
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EIGENVALUES , *BIPARTITE graphs , *MATRIX norms , *LAPLACIAN matrices , *TOPOLOGICAL degree , *GRAPH connectivity , *REGULAR graphs - Abstract
For a simple graph G, the ABC-index is a degree based topological index and is defined as where dv is the degree of the vertex υ in G. Recently, the Laplacian ABC-matrix was introduced in [22] is defined by where is the diagonal matrix of ABC-degrees and Ã(G) is the ABC-matrix of G: The eigenvalues of the matrix are called the Laplacian ABC-eigenvalues of G. In the article, we consider the problem of characterization of connected graphs having exactly three distinct Laplacian ABC-eigenvalues. We solve this problem for bipartite graphs, multipartite graphs, unicyclic graphs, regular graphs and prove the non-existence of such graphs with diameter greater than 2. We introduce the concept of trace norm of the matrix called the Laplacian ABC-energy of G. We obtain some upper and lower bounds for the Laplacian ABC-energy and characterize the extremal graphs which attain these bounds. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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