The study of chromatically unique graphs has been drawing much attention and many results are surveyed in \([4, 12, 13]\). The notion of adjoint polynomials of graphs was first introduced and applied to the study of the chromaticity of the complements of the graphs by Liu \([17]\) (see also \([4]\)). Two invariants for adjoint equivalent graphs that have been employed successfully to determine chromatic unique graphs were introduced by Liu \([17]\) and Dong et al. \([4]\) respectively. In the paper, we shall utilize, among other things, these two invariants to investigate the chromaticity of the complement of the tadpole graphs \(C_n(P_m)\), the graph obtained from a path \(P_m\) and a cycle \(C_n\) by identifying a pendant vertex of the path with a vertex of the cycle. Let \(\bar{G}\) stand for the complement of a graph \(G\). We prove the following results:

1. The graph \(\overline{{{C}_{n-1}(P_2)}}\) is chromatically unique if and only if \(n \neq 5, 7\).
2. Almost every \(\overline{{C_n(P_m)}}\) is not chromatically unique, where \(n \geq 4\) and \(m \geq 2\).

An \(L(2,1)\)-labelling of a graph \(G\) is a function \(f\) from the vertex set \(V(G)\) to the set of all nonnegative integers such that \(|f(x) – f(y)| \geq 2\) if \(d(x,y) = 1\) and \(|f(x) – f(y)| \geq 1\) if \(d(x,y) = 2\), where \(d(x,y)\) denotes the distance between \(x\) and \(y\) in \(G\). The \((2,1)\)-labelling number \(\lambda(G)\) of \(G\) is the smallest number \(k\) such that \(G\) has an \(L(2,1)\)-labelling with \(\max\{f(v) : v \in V(G)\} = k\). Griggs and Yeh conjecture that \(\lambda(G) \leq \Delta^2\) for any simple graph with maximum degree \(\Delta \geq 2\). This article considers the graphs formed by the cartesian product of \(n\) (\(n \geq 2\) graphs. The new graph satisfies the above conjecture (with minor exceptions). Moreover, we generalize our results in [19].

In this study, we first define new sequences named \((s, t)\)-Jacobsthal and \((s, t)\) Jacobsthal-Lucas sequences. After that, by using these sequences, we establish \((s, t)\)-Jacobsthal and \((s, t)\) Jacobsthal-Lucas matrix sequences. Finally, we present some important relationships between these matrix sequences.

Several transformations about \(_\gamma F_6(1)\)-series are established by applying the modified Abel lemma on summation by parts. As a consequence, a reciprocal relation on balanced \(_3F_2(1)\)-series is derived, which may also be considered as a nonterminating extension of Saalschütz’s theorem (1891).