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.

A signed total \(k\)-dominating function of a graph \(G = (V, E)\) is a function \(f: V \rightarrow \{+1, -1\}\) such that for every vertex \(v\), the sum of the values of \(f\) over the open neighborhood of \(v\) is at least \(k\). A signed total \(k\)-dominating function \(f\) is minimal if there does not exist a signed total \(k\)-dominating function \(g\), \(f \neq g\), for which \(g(v) \leq f(v)\) for every \(v \in V\).The weight of a signed total \(k\)-dominating function is \(w(f) = \sum_{v \in V} f(v)\). The signed total \(k\)-domination number of \(G\), denoted by \(\gamma_{t,k}^s(G)\), is the minimum weight of a signed total \(k\)-dominating function on \(G\).The upper signed total \(k\)-domination number \(\Gamma_{t,k}^s(G)\) of \(G\) is the maximum weight of a minimal signed total \(k\)-dominating function on \(G\).
In this paper, we present sharp lower bounds on \(\gamma_{t,k}^s(G)\) for general graphs and \(K_{r+1}\)-free graphs and characterize the extremal graphs attaining some lower bounds. Also, we give a sharp upper bound on \(\Gamma_{t,k}^s(G)\) for an arbitrary graph.

We show that a \(2\)-subset-regular self-complementary \(3\)-uniform hypergraph with \(7\) vertices exists if and only if \(n \geq 6\) and \(n\) is congruent to \(2\) modulo \(4\).

Given a graph \(G\), a function \(f: V(G) \to \{1, 2, \ldots, k\}\) is a \(k\)-ranking of \(G\) if \(f(u) = f(v)\) implies every \(u-v\)
path contains a vertex \(w\) such that \(f(w) > f(u)\). A \(k\)-ranking is minimal if the reduction of any label greater
than \(1\) violates the described ranking property.The \(arank\) number of a graph, denoted \(\psi_r(G)\),
is the maximum \(k\) such that \(G\) has a minimal \(k\)-ranking.We establish new properties for minimal rankings and present
new results for the \(arank\) number of a cycle.

In this paper, we prove that the connectivity and the edge connectivity of the lexicographic product of two graphs \(G_1\) and \(G_2\) are equal to \(\kappa_1 v_2\) and \(\min\{\lambda_1 v_2^2, \delta_2 + \delta_1v_2\}\), respectively, where \(\delta_i\), \(\kappa_i\), \(\lambda_i\), and \(n_i\) denote the minimum degree, connectivity, edge-connectivity, and number of vertices of \(G_i\), respectively.
We also obtain that the edge-connectivity of the direct product of \(K_2\) and a graph \(H\) is equal to \(\min\{2\lambda, 2\beta, \min_{j =\lambda}^\delta\{j + 2\beta_j\}\}\), where \(\theta\) is the minimum size of a subset \(F \subset E(H)\) such that \(H – F\) is bipartite and \(\beta_j = \min\{\beta(C)\}\), where \(C\) takes over all components of \(H – B\) for all edge-cuts \(B\) of size \(j \geq \lambda=\lambda (H)\).