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All Journal Electronic Journal of Graph Theory and Applications (EJGTA)
Malik, Shabnam
Forman Christian College (A Chartered University) Lahore
Electrical & Electronics Engineering

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Hamiltonicity in directed Toeplitz graphs Tn⟨1, 3, 6; t⟩ Malik, Shabnam
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 13, No 1 (2025): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2025.13.1.10

Abstract

A directed Toeplitz graph, denoted as Tn⟨s1, …, sk; t1, …, tl⟩ of order n, is a digraph in which an edge (i,  j) exists if and only if j − i = sp or i − j = tq for some 1 ≤ p ≤ k and 1 ≤ q ≤ l. The adjacency matrix of such a graph forms a Toeplitz matrix, characterized by constant values along all diagonals parallel to the main diagonal. In this paper, we explore the Hamiltonicity of directed Toeplitz graphs of the form Tn⟨1, 3, 6; t⟩. We establish that Tn⟨1, 3, 6; t⟩ is Hamiltonian for t = 5, 10 and for all t ≥ 12, for every n. Additionally, we show that the graph remains Hamiltonian for all n, with only a finite number of exceptions when t = 3, 4, 6, 7, 8, 9 and 11. Specifically, for t = 1, the graph is Hamiltonian only when n = 7, while for t = 2, it is Hamiltonian under certain conditions on n, namely when n ≡ 0, 1, 3 (mod 4). 
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