Two graphs are defined to be adjointly equivalent if their complements are chromatically equivalent. By \( h(G,x) \) and \( P(G,\lambda) \) we denote the adjoint polynomial and the chromatic polynomial of graph \( G \), respectively. A new invariant of graph \( G \), which is the fifth character \( R_5(G) \), is given in this paper. Using this invariant and the properties of the adjoint polynomials, we firstly and completely determine the adjoint equivalence class of the graph \( \zeta_n^1 \). According to the relations between \( h(G,x) \) and \( P(G,\lambda) \), we also simultaneously determine the chromatic equivalence class of \( \overline{\zeta_n^1} \).

In this paper, we discuss the properties of a class of generalized harmonic numbers \( H_{n,r} \). Using Riordan arrays and generating functions, we establish some identities involving \( H_{n,r} \). Furthermore, we investigate certain sums related to harmonic polynomials \( H_n(z) \). In particular, using the Riordan array method, we explore interesting relationships between these polynomials, the generating Stirling polynomials, the Bernoulli polynomials, and the Cauchy polynomials. Finally, we obtain the asymptotic expansion of certain sums involving \( H_{n,r} \).

We prove that \( F_v(3,5;6) = 16 \), which solves the smallest open case of vertex Folkman numbers of the form \( F_v(3, k; k+1) \). The proof uses computer algorithms.

A family \( \mathcal{G} \) of connected graphs is a family with constant metric dimension if \( \dim(G) \) is finite and does not depend upon the choice of \( G \) in \( \mathcal{G} \). The metric dimension of some classes of plane graphs has been determined in references [3], [4], [5], [12], [14], and [18], while the metric dimension of some families of convex polytopes has been studied in references [8], [9], [10], and [11]. The following open problem was raised in reference [11].

Open Problem [11]: Let \( G \) be the graph of a convex polytope which is obtained by joining the graph of two different convex polytopes \( G_1 \) and \( G_2 \) (such that the outer cycle of \( G_1 \) is the inner cycle of \( G_2 \)) both having constant metric dimension. Is it the case that \( G \) will always have constant metric dimension?

In this paper, we extend this study to an infinite class of convex polytopes obtained as a combination of the graph of an antiprism \( A_n \) [1] and the graph of convex polytope \( Q_n \) [2], such that the outer cycle of \( A_n \) is the inner cycle of \( Q_n \). It is natural to ask for the characterization of classes of convex polytopes with constant metric dimension. Note that the problem of determining whether \( \dim(G) < k \) is an NP-complete problem [7].

Let \( G \) be a finite \( 4 \)-regular cyclically \( 2k \)-edge-connected simple graph for some integer \( k \geq 1 \). Let \( E(k) \) be a set of \( k \) independent edges in \( G \) and \( (E_1, E_2) \) be a partition of \( E(k) \). We consider when there exists a \( 2 \)-factor in \( G \) which excludes all edges of \( E_1 \), and includes all the edges of \( E_2 \). A complete characterization is provided.

If an edge-disjoint decomposition of a complete graph of order \( n \) into copies of a \( 3 \)-star (i.e., the graph \( K_{1,3} \) on \( 4 \) vertices) is taken, and if these \( 3 \)-stars can be paired up in three distinct ways to form a graph on \( 6 \) vertices consisting of a \( 4 \)-cycle with two opposite pendant edges, such that:
(1) in each of the three pairings, there exists a metamorphosis into a \( 4 \)-cycle system; (2) taking precisely those \( 4 \)-cycles formed from the two pendant edges from each pair of \( 3 \)-stars, in each of the three metamorphoses, we again have a \( 4 \)-cycle system of the complete graph, then this is called a complete set of metamorphoses from paired \( 3 \)-stars into \( 4 \)-cycles.

We show that such a complete set of metamorphoses from paired \( 3 \)-stars into \( 4 \)-cycles exists if and only if the order of the complete graph is \( 1 \) or \( 9 \pmod{24} \), and greater than \( 9 \).

Let \( G \) be a connected graph of order \( n \geq 3 \) and size \( m \), and let \( f: E(G) \to \mathbb{Z}_n \) be an edge labeling of \( G \). Define an induced vertex labeling \( f’: V(G) \to \mathbb{Z}_n \) in terms of \( f \) by \( f'(v) = \sum_{u \in N(v)} f(uv) \), where the sum is computed in \( \mathbb{Z}_n \). If \( f’ \) is one-to-one, then \( f \) is called a modular edge-graceful labeling and \( G \) is a modular edge-graceful graph. It is known that no connected graph of order \( n \geq 3 \) with \( n \equiv 2 \pmod{4} \) is modular edge-graceful. A 1991 conjecture states that every tree of order \( n \) where \( n \not\equiv 2 \pmod{4} \) is modular edge-graceful. In this work, we show that this conjecture is true and furthermore that a nontrivial connected graph of order \( n \) is modular edge-graceful if and only if \( n \not\equiv 2 \pmod{4} \). The modular edge-gracefulness \(\text{meg}(G)\) of a connected graph \( G \) of order \( n \geq 3 \) is the smallest integer \( k \geq n \) for which there exists an edge labeling \( f: E(G) \to \mathbb{Z}_k \) such that the induced vertex labeling \( f’: V(G) \to \mathbb{Z}_k \) is one-to-one. It is shown that \(\text{meg}(G) = n+1\) for every connected graph \( G \) that is not modular edge-graceful.

A dominating set is a vertex subset \(D\) of a graph \(G\) such that each vertex of \(G\) is either in \(D\) or adjacent to a vertex in \(D\). The domination number, \(\gamma(G)\), is the minimum cardinality of a dominating set of a graph \(G\). In this paper, we will investigate the domination number of Fibonacci cubes. We firstly study the degree sequence of the Fibonacci cubes. Then, a lower bound for the domination number of Fibonacci cube of order \(n\) is obtained, and the exact value of the domination number of Fibonacci cubes of order at most \(8\) is determined.

Let \( L(m, n) \) be the largest integer such that, if each symbol in an \( m \times n \) rectangle occurs at most \( L(m, n) \) times, then the array must have a transversal. We improve the lower bound to \( L(m, n) \geq \left\lfloor \frac{m(n – m + 1) – 1}{m – 1} \right\rfloor \) for \( m > 1 \). Then we show that sporadically \( L(m, n) < \left\lfloor \frac{mn – 1}{m – 1} \right\rfloor \) in the range \( m \leq n \leq m^2 – 3m + 3 \). Define \( n_0(m) \) to be the smallest integer \( z \) such that if \( n \geq z \) then \( L(m, n) = \left\lfloor \frac{mn – 1}{m – 1} \right\rfloor \). We improve \( n_0(m) \) from \( O(m^3) \) to \( O(m^{2.5}) \). Finally, we determine \( L(4, n) \) for all \( n \).

In [1], we showed that for \( v \equiv 1 \) or \( 3 \pmod{6} \), there is an equitable \( k \)-edge coloring of \( K_v \) that does not admit any polychromatic \( STS(v) \), when \( k = 2, 3 \), and \( v – 2 \). In this paper, we extend the results to all feasible values of \( k \), where \( 2 \leq k \leq v – 2 \).