This paper studies the unitary Cayley graph associated with the ring of dual numbers, \(\mathbb{Z}_n[\alpha]\). It determines the exact diameter, vertex chromatic number, and edge chromatic number. In addition, it classifies all perfect graphs within this class.

Stanton-type graphs were introduced recently. In this paper, we define generalized Stanton-type graphs. We also identify LO and OE graphs, find the minimum \(\lambda\) for decomposition of \(\lambda K_n\) into these graphs, and show that for all viable values of \(\lambda\), the necessary conditions are sufficient for LO- and OE-decompositions using cyclic decompositions from base graphs.

For a finite graph \(G\) with vertices \(\{v_1, \ldots, v_r\}\), a representation of \(G\) modulo \(n\) is a set \(\{a_1, \ldots, a_r\}\) of distinct, nonnegative integers with \(0 \leq a_i < n\), satisfying \(\gcd(a_i – a_j, n) = 1\) if and only if \(v_i\) is adjacent to \(v_j\). The representation number, \(Rep(G)\), is the smallest \(n\) such that \(G\) has a representation modulo \(n\). Evans \(et \, al.\) obtained the representation number of paths. They also obtained the representation number of a cycle except for cycles of length \(2^k + 1\), \(k \geq 3\). In the present paper, we obtain upper and lower bounds for the representation number of a caterpillar, and get its exact value in some cases.

By applying the method of generating functions, the purpose of this paper is to give several summations of reciprocals related to the quintuple product of the general second-order recurrence \(\{W_{rn}\}\) for arbitrary positive integers \(r\). As applications, some identities involving Fibonacci and Lucas numbers are obtained.

A collection \(\mathcal{S}\) of proper subgroups of a group \(G\) is said to be a cover (or covering) for \(G\) if the union of the members of \(\mathcal{S}\) is all of \(G\). A cover \(\mathcal{C}\) of minimal cardinality is called a minimal cover for \(G\) and \(|\mathcal{C}|\) is called the covering number of \(G\), denoted by \(\sigma(G)\). In this paper we determine the covering numbers of the alternating groups \(A_9\) and \(A_{11}\).

Given a (not necessarily proper) coloring of a digraph \( c:V(D)\rightarrow {N}\), let \( OC(v)\) denote the set of colors assigned to the out-neighbors of \(v\). Similarly, let \( IC(v)\) denote the set of colors assigned to the in-neighbors of \(v\). Then \(c\) is a set coloring of \(D\) provided \((u,v) \in A(D)\) implies \( OC(u) \neq OC(v)\). Analogous to the set chromatic number of a graph given by Chartrand, \(et\) \(al.\) \([3]\), we define \( \chi_s(D) \) as the minimum number of colors required to produce a set coloring of \(D\). We find bounds for \(\chi_s(D)\) where \(D\) is a digraph and where \(D\) is a tournament. In addition we consider a second set coloring, where \((u,v) \in A(D)\) implies \( OC(u) \neq IC(v)\).

Let \(\mathcal{C}\) be a finite family of distinct boxes in \(\mathbb{R}^d\), with \(G\) the intersection graph of \(\mathcal{C}\), and let \(S = \cup\{C : C \in \mathcal{C}\}\). For each block of \(G\), assume that the corresponding members of \(\mathcal{C}\) have a staircase convex union. Then when \(S\) is staircase starshaped, its staircase kernel will be a staircase convex set. Moreover, this result (and others) will hold for more general families \(\mathcal{C}\) as well.

The domination chain \(\iota_r(G) \leq \gamma(G) \leq \iota(G) \leq \beta_o(G) \leq \Gamma(G) \leq IR(G)\), which holds for any graph \(G\), is the subject of much research. In this paper, we consider the maximum number of edges in a graph having one of these domination chain parameters equal to \(2\) through a unique realization. We show that a specialization of the domination chain still holds in this setting.

In this paper, we study further bounds of constant dimension codes in Grassmannian space \(\mathcal{G}_q(n,k)\). There is increasing interest in subspace codes since they are essential for error-correction in networks. Additionally, there is a connection to the theory over finite fields. By revising the specific construction methods of the constant dimension codes in [1], [2], we improve some bounds on \(q\)-ary constant dimension codes in certain cases.

In this paper, we use a recent result of Bryant, Horsley, and Pettersson in [1] to provide an alternate and more straightforward proof of results concerning neighborhood graphs in maximum packings of \(2K_n\) with triples, some of which were only recently obtained.

To set the stage, consider any partial triple system \((V,B)\) of \(2K_n\). In this system, the neighborhood of a vertex \(v\) is defined as the subgraph induced by the set \(\{\{x,y\} \mid \{v,x,y\} \in B\}\). This concept plays a crucial role in the results initially obtained by Colbourn and Rosa for \(n \equiv 0,1 \pmod{3}\) and by Chaffee and Rodger for \(n \equiv 2 \pmod{3}\). These results offer a complete characterization of the possible neighborhoods in a maximum packing of \(2K_n\).

In both of these original papers, the authors employed difference methods—a combinatorial technique that often involves selecting pairs of elements from a group and studying their differences—and a pull-up technique, which is used to modify the neighborhood of a vertex. However, despite the effectiveness of these methods, neither approach seems to lend itself easily to deriving the results of the other.

In our paper, we present a more unified and simplified proof that brings both of these results together. By leveraging the recent findings of Bryant, Horsley, and Pettersson, we can bypass the need for the more complex difference methods and pull-up techniques, instead relying on the underlying principles elucidated in their work. This approach not only simplifies the proof process but also provides a clearer and more direct route to understanding the structure of neighborhood graphs in these maximum packings.