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.

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.