The spectral radius of a graph is the largest eigenvalue of its adjacency matrix. Let \(\mathcal{G}\) be the set of unicyclic graphs of order \(n\) with girth \(g\). For all integers \(n\) and \(g\) with \(5 \leq g \leq n – 6\), we determine the first \(|\frac{g}{2}| + 3\) spectral radii of unicyclic graphs in the set \(\mathcal{U}_n^g\).

In this paper, we consider labelings of graphs in which the label on an edge is the absolute value of the difference of its vertex labels. Such a labeling using \(\{0,1,2,\ldots,k-1\}\) is called \(k\)-equitable if the number of vertices (resp. edges) labeled \(i\) and the number of vertices (resp. edges) labeled \(j\) differ by at most one and is called \(k\)-balanced if the number of vertices labeled \(i\) and the number of edges labeled \(j\) differ by at most one. We determine which graphs in certain families are \(k\)-equitable or \(k\)-balanced and we give also some necessary conditions on these two labelings.

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).

An almost-bipartite graph is a non-bipartite graph with the property that the removal of a particular single edge renders the graph bipartite. A graph labeling of an almost-bipartite graph \(G\) with \(n\) edges that yields cyclic \(G\)-decompositions of the complete graph \(K_{2nt+1}\) was recently introduced by Blinco, El-Zanati, and Vanden Eynden. They called such a labeling a \(\gamma\)-labeling. Here we show that the class of almost-bipartite graphs obtained from a path with at least \(3\) edges by adding an edge joining distinct vertices of the path an even distance apart has a \(\gamma\)-labeling.

The locally twisted cube \(LTQ_n\) is an important variation of hypercube and possesses many desirable properties for interconnection networks. In this paper, we investigate the problem of embedding paths in faulty locally twisted cubes. We prove that a path of length \(l\) can be embedded between any two distinct vertices in \(LTQ_n – F\) for any faulty set \(F \subseteq V(LTQ_n) \cup E(LTQ_n)\) with \(|F| \leq n-3\) and any integer \(l\) with \(2^{n-1} \leq l \leq |V(LTQ_n – F)| – 1\) for any integer \(n > 3\). The result is tight with respect to the two bounds on path length \(l\) and faulty set size \(|F|\) for a successful embedding.

A \(G\)-design is a partition of \(E(K_v)\) in which each element induces a copy of \(G\). The existence of \(G\)-designs with the additional property that they contain no proper subsystems has been previously settled when \(G \in \{K_3, K_4 – e\}\). In this paper, the existence of \(P_m\)-designs which contain no proper subsystems is completely settled for every value of \(m\) and \(v\).

The Randić index of an organic molecule whose molecular graph is \(G\) is the sum of the weights \((d(u)d(v))^{-\frac{1}{2}}\) of all edges \(uv\) of \(G\), where \(d(u)\) and \(d(v)\) are the degrees of the vertices \(u\) and \(v\) in \(G\). In this paper, we give a sharp lower bound on the Randić index of cacti with perfect matchings.