In this study, we investigate Diophantine equations using the generalized Fibonacci and Lucas sequences. We obtain all integer solutions for several Diophantine equations such as \(x^2 -kxy- y^2 = \mp 1,\) \(x^2 -kxy+ y^2 = 1,\) \(x^2 – kxy-y^2 = \mp (k^2+4),\)
\(x^2 – (k^2 + 4)xy + (k^2+4)y^2 =\mp k^2,\) \(x^2 – kxy +y^2 = -(k^2-4)\). and \(x^2-(k^2-4)xy-(k^2-4)y^2=k^2\)
Some of these results are previously known, but we provide new and distinct proofs using generalized Fibonacci and Lucas sequences.

Let \(G = \{g_1, \ldots, g_n\}\) be a finite abelian group. Consider the complete graph \(K_n\) with vertex set \(\{g_1, \ldots, g_n\}\). A \(G\)-coloring of \(K_n\) is a proper edge coloring where the color of edge \(\{g_i, g_j\}\) is \(g_i + g_j\), \(1 \leq i 2\), there exists a proper edge coloring of \(K_p\) which is decomposable into multicolored Hamilton cycles.

It is shown that \(r(K_{1,m,k}, K_n) \leq (k – 1 + o(1)) (\frac{n}{log n})^{m+1}\) for any two fixed integers \(k \geq m \geq 2\) and \(n \to \infty\).
This result is obtained using the analytic method and the function \(f_{m}(x) = \int_0^1 \frac{(1-t)^{\frac{1}{m}}dt}{m+(x-m)^t} , \quad x \geq 0,m \geq 1,\)
building upon the upper bounds for \(r(K_{m,k}, K_n)\) established by Y. Li and W. Zang.Furthermore, \((c – o(1)) (\frac{n}{log n})^{\frac{7}{3}}\leq r(W_{4}, K_n) \leq (1 + o(1)) (\frac{n}{log n})^{3}\) (as \(n \to \infty\)). Moreover, we derive
\(r(K_{1} + K_{m,k}, K_n) \leq (k – 1 + o(1)) (\frac{n}{log n})^{l+m}\) for any two fixed integers \(k \geq m \geq 2\) (as \(n \to \infty\)).

A simple graph \(G = (V, E)\) admits an \(H\)-covering if every edge in \(E\) belongs to a subgraph of \(G\) isomorphic to \(H\). We say that \(G\) is \(H\)-magic if there exists a total labeling \(f: V \cup E \rightarrow \{1, 2, \ldots, |V| + |E| + 1\}\) such that for each subgraph \(H’ = (V’, E”)\) of \(G\) isomorphic to \(H\),
\(\sum_{v \in V’} f(v) + \sum_{e \in E”} f(e)\)
is constant.

When \(f(V) = \{1, 2, \ldots, |V|\}\), then \(G\) is said to be \(H\)-supermagic.

In this paper, we show that all prism graphs \(C_n \times P_m\), except for \(n = 4\), the ladder graph \(P_3 \times P_n\), and the grid \(P_3 \times P_n\), are \(C_4\)-supermagic.

The average crosscap number of a graph \(G\) is the expected value of the crosscap number random variable, over all labeled \(2\)-cell non-orientable embeddings of \(G\). In this study, some experimental results for average crosscap number are obtained. We calculate all average crosscap numbers of graphs with Betti number less than \(5\). As a special case, the smallest ten values of average crosscap number are determined. The distribution of average crosscap numbers of all graphs in \({R}\) is sparse. Some structure theorems for average crosscap number with a given or bounded value are provided. The exact values of average crosscap numbers of cacti and necklaces are determined. The crosscap number distributions of cacti and necklaces of type \((r,0)\) are proved to be strongly unimodal, and the mode of the embedding distribution sequence is upper-rounding or lower-rounding of its average crosscap number. Some open problems are also proposed.

A Roman dominating function of a graph \(G\) is a labeling \(f: V(G) \rightarrow \{0,1,2\}\) such that every vertex with label \(0\) has a neighbor with label \(2\). The Roman domination number \(\gamma_R(G)\) of \(G\) is the minimum of \(\sum_{v \in V(G)} f(v)\) over such functions. The Roman domination subdivision number \(sd_{\gamma R}(G)\) is the minimum number of edges that must be subdivided (each edge in \(G\) can be subdivided at most once) in order to increase the Roman domination number.

In this paper, we prove that if \(G\) is a graph of order \(n \geq 4\) such that \(\overline{G}\) and \(G\) have connected components of order at least \(3\), then
\(sd_{\gamma R}(G) + sd_{\gamma R}(\overline{G}) \leq \left\lfloor \frac{n}{2} \right\rfloor + 3.\)

In \textit{Ars Comb.} \({84} (2007), 85-96\), Pedersen and Vestergaard posed the problem of determining a lower bound for the number of independent sets in a tree of fixed order and diameter \(d\). Asymptotically, we give here a complete solution for trees of diameter \(d \leq 5\). The lower bound is \(5^{\frac{n}{3}}\) and we give the structure of the extremal trees. A generalization to connected graphs is stated.

Let \(\mathcal{G}\) be a family of graphs. The anti-Ramsey number \(\text{AR}(n, \mathcal{G})\) for \(\mathcal{G}\) is the maximum number
of colors in an edge coloring of \(K_n\) that has no rainbow copy of
any graph in \(\mathcal{G}\). In this paper, we determine the bipartite anti-Ramsey number for the family of trees with
\(k\) edges.

Let \(G\) be a finite group of order \(n\) and \(S\) (possibly containing the identity element) be a subset of \(G\). The Bi-Cayley graph
\(\text{BC}(G, S)\) of \(G\) is a bipartite graph with vertex set \(G \times \{0, 1\}\) and edge set \(\{(g, 0), (gs, 1) \mid g \in G, s \in S\}\). Let \(p\) (\(0 < p < 1\)) be a fixed number.We define \({B} = \{\text{BC}(G, S) \mid S \subseteq G\}\) as a sample space and assign a probability measure by requiring \(P_r(X) = p^k q^{n-k}\) for \(X = \text{BC}(G, S)\) with \(|S| = k\), where \(q = 1-p\). It is shown that the probability of the set of Bi-Cayley graphs of \(G\) with diameter \(3\) approaches \(1\) as the order \(n\) of \(G\) approaches infinity.

In this study, we define and investigate the Gaussian Jacobsthal and Gaussian Jacobsthal Lucas numbers. We derive generating functions, Binet formulas, explicit formulas, and matrix representations for these numbers. Additionally, we present explicit combinatorial and determinantal expressions, examine negatively subscripted numbers, and establish various identities. Our results parallel those for the Jacobsthal and Jacobsthal Lucas numbers, yielding interesting consequences for the Gaussian Jacobsthal and Gaussian Jacobsthal Lucas numbers.