Let \(G_i\) be the subgraph of \(G\) whose edges are in the \(i\)-th color in an \(r\)-coloring of the edges of \(G\). If there exists an \(r\)-coloring of the edges of \(G\) such that \(H_i \cong G_i\) for all \(1 \leq i \leq r\), then \(G\) is said to be \(r\)-colorable to \((H_1, H_2, \ldots, H_r)\). The multicolor Ramsey number \(R(H_1, H_2, \ldots, H_r)\) is the smallest integer \(n\) such that \(K_n\) is not \(r\)-colorable to \((H_1, H_2, \ldots, H_r)\). Let \(C_m\) be a cycle of length \(m\). The four-color Ramsey numbers related to \(C_6\) are studied in this paper. It is well known that \(18 \leq R_4( C_6) \leq 21\). We prove that \(R(C_5, C_4, C_4, C_4) = 19\) and \(18 \leq R(C_6, C_6, H_1, H_2) \leq 20\), where \(H_i\) are isomorphic to \(C_4\) or \(C_6\).

A graph \(G\) is called an \(M_r(k)\)-graph if \(G\) has no \(k\)-list assignment to its vertices with exactly \(r\) vertex colorings. We characterize all \(M_3(2)\)-graphs. More precisely, it is shown that a connected graph \(G\) is an \(M_3(2)\)-graph if and only if each block of \(G\) is a complete graph with at least three vertices.

A global boundary defensive \(k\)-alliance in a graph \(G = (V, E)\) is a dominating set \(S\) of vertices of \(G\) with the property that every vertex in \(S\) has \(\geq k\) more neighbors in \(S\) than it has outside of \(S\). A global boundary offensive \(k\)-alliance in a graph \(G\) is a set \(S\) of vertices of \(G\) with the property that every vertex in \(V \setminus S\) has \(k\) more neighbors in \(S\) than it has outside of \(S\). We define a global boundary powerful \(k\)-alliance as a set \(S\) of vertices of \(G\), which is both global boundary defensive \(k\)-alliance and global boundary offensive \((k+2)\)-alliance. In this paper, we study mathematical properties of boundary powerful \(k\)-alliances. In particular, we obtain several bounds (closed formulas for the case of regular graphs) on the cardinality of every global boundary powerful \(k\)-alliance. Additionally, we consider the case in which the vertex set of a graph \(G\) can be partitioned into two boundary powerful \(k\)-alliances, showing that, in such a case, \(k = -1\) and, if \(G\) is \(\delta\)-regular, its algebraic connectivity is equal to \(\delta + 1\).

We present two recursive enumeration formulas for the number of labelled essential graphs. The enumeration parameters of the first formula are the number of vertices, chain components, and cliques, while the enumeration parameters of the second formula are the number of vertices and cliques.Both formulas may be used to count the number of labelled essential graphs
with given number of vertices.

In this paper, we first survey the connections between Bell polynomials (numbers) and the derangement polynomials (numbers). Their close relations are mainly based on Hsu’ summation formula. According to this formula, we present some new identities involving harmonic numbers,Bell polynomials (numbers) and the derangement polynomials (numbers).Moreover, we find that the series \(\sum_{m\geq0}(\frac{D_m}{m!}-\frac{1}{e})\) is (absolutely) convergent and their sums are also determined, where \(D_m\) is the \(mth\) derangement number.

A graph \(G\) is regular if the degree of each vertex of \(G\) is d and almost regular or more precisely a \((d,d + 1)\)-graph, if the degree of each vertex of \(G\) is either \(d\) or \(d+1\). If \(d \geq 2\) is an integer, \(G\) a triangle-free \((d,d + 1)\)-graph of order n without an odd component and \(n \leq 4d\), then we show in this paper that \(G\) contains a perfect matching. Using a new Turdn type result, we present an analogue for triangle-free regular graphs. With respect to these results, we construct smallest connected, regular and almost regular triangle-free even order graphs without perfect matchings.

In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski developed a graph transformation that transforms a graph \(G\) into a new graph \(\mu(G)\), which is called the Mycielskian of \(G\).This paper shows that:
For a strongly connected digraph \(D\) with \(|V(D)| \geq 2\):\(\mu(D)\) is super-\(\kappa\) if and only if \(\delta(D) < 2\kappa(D)\).;\(\mu(D)\) is super-\(\lambda\) if and only if \(D \ncong \overrightarrow{K_2}\).

The sum of the squares of eccentricity \((SSE)\) over all vertices of a connected graph is a new graph invariant proposed in \([13]\) and further studied in \([14, 15]\). In this paper, we report some further mathematical properties of \(SSE\). We give sharp lower bounds for \(SSE\) among all \(n\)-vertices connected graphs with given independence number, vertex-, and edge-connectivity, respectively. Addtionally, we give explicit formulas for \(SSE\) of Cartesian product of two graphs, from which we deduce \(SSE\) of \(C_4\), nanotube and nanotorus.

The vertex linear arboricity \(vla(G)\) of a nonempty graph \(G\) is the minimum number of subsets into which the vertex set \(V(G)\) can be partitioned so that each subset induces a subgraph whose connected components are paths.An integer distance graph is a graph \(G(D)\) with the set of all integers as vertex set and two vertices \(u,v \in {Z}\) are adjacent if and only if \(|u-v| \in D\), where the distance set \(D\) is a subset of the positive integers.Let \(D_{m,k,3} = [1,m] \setminus \{k, 2k, 3k\}\) for \(m \geq 4k \geq 4\). In this paper, we obtain some upper and lower bounds of the vertex linear arboricity of the integer distance graph \(G(D_{m,k,3})\) and the exact value of it for some special cases.

In this paper, we generalize to the class of signed graphs the well known result that every numbered graph can be embedded as an induced subgraph in a gracefully numbered graph.

There are \(267\) nonisomorphic groups of order \(64\). It was known that \(259\) of these groups admit \((64, 28, 12)\) difference sets and the other eight groups do not admit \((64, 28, 12)\) difference sets. Despite this result, no research investigates the problem of finding all \((64, 28, 12)\) difference sets in a certain group of order \(64\).In this paper, we find all \((64, 28, 12)\) difference sets in \(111\) groups of order \(64. 106\) of these groups are nonabelian. The other five are \(\mathbb{Z}_{16} \times \mathbb{Z}_4\), \(\mathbb{Z}_{16} \times \mathbb{Z}_2^2\), \(\mathbb{Z}_8 \times \mathbb{Z}_8\), \(\mathbb{Z}_8 \times \mathbb{Z}_4 \times \mathbb{Z}_2\), and \(\mathbb{Z}_8 \times \mathbb{Z}_2^3\).In these \(111\) groups, we obtain \(74,922\) non-equivalent \((64, 28, 12)\) difference sets. These difference sets provide at least \(105\) nonisomorphic symmetric \((64, 28, 12)\) designs. Most of our work was done using programs with the software \(GAP\).

In this paper, we obtain some generating functions for the generalized Zernike or disk polynomials \(P_{m,n}^\alpha (z,z^*)\) which are investigated by Wiinsche [13]. We derive various families of bilinear and bilateral generating functions. Furthermore, some special cases of the results presented in this study are indicated. Also, it is possible to obtain multilinear and multilateral generating functions for the polynomials \(P_{m,n}^\alpha (z,z^*)\).

A \((k,t)\)-list assignment \(L\) of a graph \(G\) is a list of \(k\) colors available at each vertex \(v\) in \(G\) such that \(|\bigcup_{v\in V(G)}L(v)| = t\). A proper coloring \(c\) such that \(c(v) \in L(v)\) for each \(v \in V(G)\) is said to be an \(L\)-coloring. We say that a graph \(G\) is \(L\)-colorable if \(G\) has an \(L\)-coloring. A graph \(G\) is \((k,t)\)-choosable if \(G\) is \(L\)-colorable for every \((k,t)\)-list assignment \(L\).
Let \(G\) be a graph with \(n\) vertices and \(G\) does not contain \(C_5\) or \(K_{k-2}\) and \(K_{k+1}\). We prove that \(G\) is \((k, kn – k^2 – 2k)\)-choosable for \(k \geq 3\).\(G\) is not \((k, kn – k^2 – 2k)\)-choosable for \(k = 2\).This result solves a conjecture posed by Chareonpanitseri, Punnim, and Uiyyasathian [W. Chareonpan-itseri, N. Punnim, C. Uiyyasathian, On \((k,t)\)-choosability of Graphs: Ars Combinatoria., \(99, (2011) 321-333]\).

We call a graph \(G\) a \({generalized \;split \; graph}\) if there exists a core \(K\) of \(G\) such that \(V(G) \setminus V(K)\) is an independent set of \(G\).Let \(G\) be a generalized split graph with a partition \(V(G) = K \cup S\), where \(K\) is a core of \(G\) and \(S\) is an independent set. We prove that \(G\) is end-regular if and only if for any \(a, b \in S\), \(\phi \in \text{Aut}(K)\), the inclusion \(\phi(N(a)) \subsetneqq N(b)\) does not hold.
\(G\) is end-orthodox if and only if \(G\) is end-regular and for any \(a, b \in S\), \(N(a) \neq N(b)\).

In this paper we generalize the Fibonacci numbers and the Lucas numbers with respect to \(n\), respectively \(n+1\) parameters. Using these definitions we count special subfamilies of the set of \(n\) integers. Next we give the graph interpretations of these numbers with respect to the number of \(P_k\),-matchings in special graphs and we apply it for proving some identity and also for counting other subfamilies of the set of n integers.

The Wiener-Hosoya index was firstly introduced by M. Randié¢ in \(2004\). For any tree \(T\), the Wiener-Hosoya index is defined as

\[WH(T)= \sum\limits_{e\in E(T)} (h(e) + h[e])\]

where \(e = uv\) is an arbitrary edge of \(T\), and \(h(e)\) is the product of the numbers of the vertices in each component of \(T – e\), and \(h[e]\) is the product of the numbers of the vertices in each component of \(T- \{u,v\}\). We shall investigate the Wiener-Hosoya index of trees with diameter not larger than \(4\), and characterize the extremal graphs in this paper.

Our paper deals about identities involving Bell polynomials. Some identities on Bell polynomials derived using generating function and
successive derivatives of binomial type sequences. We give some relations between Bell polynomials and binomial type sequences in
first part, and, we generalize the results obtained in \([4]\) in second part.

Fouquet and Jolivet conjectured that if \(G\) is a \(k\)-connected \(n\)-vertex graph with independence number \(\alpha \geq k \geq 2\), then \(G\) has circumference at least \( \frac{k(n+\alpha-k)}{\alpha} \). This conjecture was recently proved by \(O\), West, and Wu.
In this note, we consider the set of \(k\)-connected \(n\)-vertex graphs with independence number \(\alpha > k \geq 2\) and circumference exactly \( \frac{k(n+\alpha-k)}{\alpha} \). We show that all of these graphs have a similar structure.

Let \(\Gamma\) be the rank three \(M_{24}\) maximal \(2\)-local geometry. For the two conjugacy types of involution in \(M_{24}\), we describe the fixed point sets of chambers in \(\Gamma\).

In this paper, all connected graphs with the fourth largest signless-Laplacian eigenvalue less than two are determined.

The Lights Out game on a graph \(G\) is played as follows. Begin with a (not necessarily proper) coloring of \(V(G)\) with elements of \(\mathbb{Z}_2\). When a vertex is toggled, that vertex and all adjacent vertices change their colors from \(0\) to \(1\) or vice-versa. The game is won when all vertices have color \(0\). The winnability of this game is related to the existence of a parity dominating set.
We generalize this game to \(\mathbb{Z}_k\), \(k \geq 2\), and use this to define a generalization of parity dominating sets. We determine all paths, cycles, and complete bipartite graphs in which the game over \(\mathbb{Z}_k\) can be won regardless of the initial coloring, and we determine a constructive method for creating all caterpillar graphs in which the Lights Out game cannot always be won.

A total coloring of a simple graph \(G\) is a coloring of both the edges and the vertices. A total coloring is proper if no two adjacent or incident elements receive the same color.The minimum number of colors required for a proper total coloring of \(G\) is called the total chromatic number of \(G\) and denoted by \(\chi_t(G)\). The Total Coloring Conjecture (TCC) states that for every simple graph \(G\),\(\Delta(G) + 1 \leq \chi_t(G) \leq \Delta(G) + 2.\) \(G\) is called Type \(1\) (resp. Type \(2\)) if \(\chi_t(G) = \Delta(G) +1\) (resp. \(\chi_t(G) = \Delta(G) + 2\)). In this paper, we prove that the folded hypercubes \(FQ_n\), is of Type \(1\) when \(n \geq 4\).

Let \(H\) be a simple graph with \(n\) vertices and \(\mathcal{G} = \{G_1, G_2, \ldots, G_n\}\) be a sequence of \(n\) rooted graphs.
Following Godsil and McKay (Bull. Austral. Math. Soc. \(18 (1978) 21-28\)) defined the the rooted product \(H({G})\) of \(H\) by \({G}\) is defined by identifying the root of \(G_i\) with the \(i\)th vertex of \(H\).In this paper, we calculate the Wiener index of \(H({G})\), i.e., the sum of distances between all pairs of vertices, in terms of the Wiener indices of \(G_i\), \(i = 1, 2, \ldots, k\).As an application, we derive a recursive relation for computing the Wiener index of Generalized Bethe trees.

Let \(G\) be a connected graph with \(p\) vertices and \(q\) edges.A \(\gamma\)-labeling of \(G\) is a one-to-one function f from \(V(G)\) to \({0,1,…,q}\) that induces a labeling \(f’\) from \(V(G)\) to \({1,2,…,q}\) defined by \(f(e) = |f(u) – f(v)|\) for each edge \(e = uv\) of \(G\). The value of a \(\gamma\)-labeling \(f\) is defined to be the sum of the values of \(f’\) over all
edges. Also, the maximum value of a \(\gamma\)-labeling of \(G\) is defined as the maximum of the values among all \(\gamma\)-labelings of \(G,\) while the minimum value is the minimum of the values among all \(\gamma\)-labelings
of \(G\). In this paper, the maximum value and minimum value are determined for any complete bipartite graph.

A labeling f of a graph G is a bijection from its edge set \(E(G)\) to the set \(\{1, 2, …, |E(G)|\}\), which is antimagic if for any distinct vertices \(x\) and \(y\), the sum of the labels on edges incident to \(x\) is different from the sum of the labels on edges incident to \(y\). A graph G is antimagic if \(G\) has an f which is antimagic. Hartsfield and Ringel conjectured in \(1990\)
that every connected graph other than Ko is antimagic. In this paper, we show that some graphs with regular subgraphs are antimagic.

In \([1]\), the author provided a Gray code for the set of \(n\)-length permutations with a given number of left-to-right minima in in-version array representation. In this paper, we give the first Gray code for the set of \(n\)-length permutations with a given number of left-to-right minima in one-line representation. In this code, each permutation is transformed into its successor by a product with a transposition or a cycle of length three. Also a generating algorithm for this code is given.

We introduce a generalization of the well-known concept of graceful labeling. Given a graph \(\Gamma\) with \(e = d.m\) edges, we define a \(d\)-graceful labeling of \(G\) as an injective function \(f: V(G) \rightarrow \{0, 1, 2, \ldots, d(m+1) – 1\}\) such that \(\{|f(x) – f(y)| : \{x, y\} \in E(\Gamma)\}\) = \(\{1, 2, 3, \ldots, d(m+1) – 1\} – \{m+1, 2(m+1), \ldots, (d-1)(m+1)\}.\) In the case of \(d = 1\) and of \(d = e\) we find the classical notion of a graceful labeling and of an odd graceful labeling, respectively.Also, we call \(d\)-graceful \(\alpha\)-labeling of a bipartite graph \(\Gamma\) a \(d\)-graceful labeling of \(\Gamma\) with the property that its maximum value on one of the two bipartite sets does not reach its minimum value on the other
one. We show that these new concepts allow to obtain certain cyclic graph decompositions. We investigate the existence of \(d\)-graceful \(\alpha\)-labelings for several classes of bipartite graphs, completely solving the problem for paths and stars and giving partial results about cycles of even length and ladders.

Given a graph \(H\), a graphic sequence \(\pi = (d_1, d_2, \ldots, d_n)\) is said to be potentially \(H\)-graphic if there exists a realization of \(\pi\) containing \(H\) as a subgraph.In this paper, we characterize potentially \(K_6 – E(K_3)\)-graphic sequences without zero terms, where \(K_6 – E(K_3)\) denotes the graph obtained from a complete graph on \(6\) vertices by deleting three edges forming a triangle.This characterization implies the value of \(\sigma(K_6 – E(K_3), n)\).

We propose and study game-theoretic versions of independence
in graphs. The games are played by two players – the aggressor and the
defender – taking alternate moves on a graph G with tokens located on
vertices from an independent set of \(G\). A move of the aggressor is to select
a vertex v of \(G\). A move of the defender is to move tokens located on
vertices in \(N_G(v)\) each along one incident edge. The goal of the defender is
to maintain the set of occupied vertices independent while the goal of the
aggressor is to make this impossible. We consider the maximum number of
tokens for which the aggressor can not win in a strategic and an adaptive
version of the game.

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.

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 \(j \geq \lambda=\lambda (H)\).