An edge irregular total \(k\)-labeling of a graph \(G = (V, E)\) is a labeling \(f: V \cup E \to \{1, 2, \ldots, k\}\) such that the total edge-weights \(wt(xy) = f(x) + f(xy) + f(y)\) are distinct for all pairs of distinct edges. The minimum \(k\) for which \(G\) has an edge irregular total \(k\)-labeling is called the total edge irregularity strength of \(G\). In this paper, we determine the exact value of the total edge irregularity strength of the Cartesian product of two paths \(P_n\) and \(P_m\). Our result provides further evidence supporting a recent conjecture of Ivančo and Jendrol.

For a vertex set \(S\) with cardinality at least \(2\) in a graph \(G\), a tree connecting \(S\), known as a Steiner tree or \(S\)-tree, is required. Two \(S\)-trees \(T\) and \(T’\) are internally disjoint if \(V(T) \cap V(T’) = S\) and \(E(T) \cap E(T’) = \emptyset\). Let \(\kappa_G(G)\) denote the maximum number of internally disjoint Steiner trees connecting \(S\) in \(G\). The generalized \(k\)-connectivity \(\kappa_k(G)\) of \(G\), introduced by Chartrand et al., is defined as \(\min_{S \subseteq V(G), |S|=k} \kappa_G(S)\). This paper establishes a sharp upper bound for generalized \(k\)-connectivity. Furthermore, graphs of order \(n\) with \(\kappa_3(G) = n-2,n-3\) are characterized.

A hypergraph \(\mathcal{H}\) is said to be \(p\)-Helly when every \(p\)-wise intersecting partial hypergraph \(\mathcal{H}’\) of \(H\) has nonempty total intersection. Such hypergraphs were characterized by Berge and Duchet in 1975, and since then they have appeared in various contexts, particularly for \(p=2\), where they are known as Helly hypergraphs. An interesting generalization due to Voloshin considers both the number of intersecting sets and their intersection sizes: a hypergraph \(\mathcal{H}\) is \((p,q,s)\)-Helly if every \(p\)-wise \(q\)-intersecting partial hypergraph \(\mathcal{H}’\) of \(H\) has total intersection of cardinality at least \(s\). This work proposes a characterization for \((p,q,s)\)-Helly hypergraphs, leading to an efficient algorithm for recognizing such hypergraphs when \(p\) and \(q\) are fixed parameters.

A \(k\)-chromatic graph \(G\) is \(uniquely\) \(k\)-\(colorable\) if \(G\) has only one \(k\)-coloring up to permutation of the colors. In this paper, we focus on uniquely \(k\)-colorable graphs on surfaces. Let \({F}^2\) be a closed surface, excluding the sphere, and let \(\chi({F}^2)\) denote the maximum chromatic number of graphs embeddable on \({F}^2\). We shall prove that the number of uniquely \(k\)-colorable graphs on \({F}^2\) is finite if \(k \geq 5\), and characterize uniquely \(\chi({F}^2)\)-colorable graphs on \({F}^2\). Moreover, we completely determine uniquely \(k\)-colorable graphs on the projective plane for \(k \geq 5\).

Given a distribution \(D\) of pebbles on the vertices of a graph \(G\), a pebbling move consists of removing two pebbles from a vertex and placing one on an adjacent vertex (the other is discarded). The pebbling number of a graph, denoted by \(f(G)\), is the minimal integer \(k\) such that any distribution of \(k\) pebbles on \(G\) allows one pebble to be moved to any specified vertex by a sequence of pebbling moves. In this paper, we calculate the pebbling number of the graph \(D_{n,C_m}\) and consider the relationship the pebbling number between the graph \(D_{n,C_m}\) and the subgraphs of \(D_{n,C_m}\).

Let \(G\) and \(H\) be two graphs. A proper vertex coloring of \(G\) is called a dynamic coloring if, for every vertex \(v\) with degree at least \(2\), the neighbors of \(v\) receive at least two different colors. The smallest integer \(k\) such that \(G\) has a dynamic coloring with \(k\) colors is denoted by \(\chi_2(G)\). We denote the Cartesian product of \(G\) and \(H\) by \(G \square H\). In this paper, we prove that if \(G\) and \(H\) are two graphs and \(\delta(G) \geq 2\), then \(\chi_2(G \square H) \leq \max(\chi_2(G), \chi(H))\). We show that for every two natural numbers \(m\) and \(n\), \(m, n \geq 2\), \(\chi_2(P_m \square P_n) = 4\). Additionally, among other results, it is shown that if \(3\mid mn\), then \(\chi_2(C_m \square C_n) = 3\), and otherwise \(\chi_2(C_m \square C_n) = 4\).

In \([1]\), Hosam Abdo and Darko Dimitrov introduced the total irregularity of a graph. For a graph \(G\), it is defined as
\[\text{irr}_t(G) =\frac{1}{2} \sum_{{u,v} \in V(G)} |d_G(u) – d_G(v)|,\]
where \(d_G(u)\) denotes the vertex degree of a vertex \(u \in V(G)\). In this paper, we introduce two transformations to study the total irregularity of unicyclic graphs and determine the graph with the maximal total irregularity among all unicyclic graphs with \(n\) vertices.

We consider a variation on the Tennis Ball Problem studied by Mallows-Shapiro and Merlini, \(et \;al\). The solution to the original problem is the well known Catalan numbers, while the variations discussed in this paper yield the Motzkin numbers and other related sequences. For this variation, we present a generating function for the sum of the labels on the balls.

A graph \(G\) of order \(n\) is called a tricyclic graph if \(G\) is connected and the number of edges of \(G\) is \(n + 2\). Let \(\mathcal{T}_n\) denote the set of all tricyclic graphs on \(n\) vertices. In this paper, we determine the first to nineteenth largest Laplacian spectral radii among all graphs in the class \(\mathcal{T}_n\) (for \(n \geq 11\)), together with the corresponding graphs.

The Hosoya index of a graph is defined as the total number of the matchings of the graph. In this paper, we determine the lower bounds for the Hosoya index of unicyclic graph with a given diameter. The corresponding extrenal graphs are characterized.