We show that some well-known integer sequences can be represented as sequences of determinant of matrices associated with certain families of threshold graphs. Specifically, let \(G_n\) denote a connected threshold graph with n vertices. Define \(S(G_n):=(s_{ij})\) as the \(n \times n\) matrix, where \(s_{ii}=0\) for all i; \(s_{ij}=1\) if vertices i and j are not adjacent, and \(-1\) if i and j are adjacent. We show that if \(\{\mu _{j-1}\}_{j \ge 1}\) is the sequence of Pell numbers, then \(\mu _{j-1}=|\det \,S(A_j)|\) , where \(A_j\) is the connected antiregular graph on j vertices. We then find a sequence of connected threshold graphs \(\{G_m\}\) such that the sequence of Fibonacci numbers aligns with \(\{|\det \, S(G_m)|\}\) . Additionally, we determine a recurrence relation for the sequence \(\{\det \, D(A_k)\}\) , where \(D(A_k)\) is the distance matrix of the connected antiregular threshold graph with k vertices. Using this relation, we give an explicit formula to compute \(\det \, D(A_k)\) .
