1. On the computation of Seidel Laplacian eigenvalues for graph-based binary codes.
- Author
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Mesnager, Sihem, Raja, Rameez, and Wagay, Samir Ahmad
- Subjects
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BINARY codes , *EIGENVALUES , *DIVISOR theory , *POLYNOMIAL rings , *REGULAR graphs , *INTEGERS - Abstract
Let G = (V , E) be a simple graph with vertex set as V and edge set as E. Within the framework of graph-based binary coding, a graph G is a threshold graph if and only if it can be generated by a binary code of the type 0 s 1 1 t 1 ... 0 s k 1 t k , where s i and t i are positive integers with 1 ≤ i ≤ k. This type of binary code is called a binary generating code of G. In this paper, we address a problem originally posed by Harary and Schwenk in [10] titled "Which graphs have integral spectra?" in which the main focus of our investigations lies in the Seidel Laplacian spectrum of G generated by 0 s 1 1 t 1 ... 0 s k 1 t k . We establish that all eigenvalues of G are integers and multiplicities of eigenvalues are equal to powers of bits 0 and 1 (sizes of strings of 0's and 1's) involved in a binary generating code of G. We exhibit a necessary and sufficient condition in terms of eigenvalues of the matrix associated with a graph-based G binary code to be Seidel Laplacian integral. Furthermore, for any prime p , we explore the Seidel Laplacian spectrum of zero-divisor graphs associated with the ring Z p m (where m is a positive integer) and the polynomial ring Z p [ x ] / (x p). [ABSTRACT FROM AUTHOR]
- Published
- 2024
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