Let \(\mathbb{F}_q^n\) denote the \(n\)-dimensional row vector space over the finite field \(\mathbb{F}_q\), where \(n \geq 2\). An \(l\)-partial linear map of \(\mathbb{F}_q^n\) is a pair \((V, f)\), where \(V\) is an \(l\)-dimensional subspace of \(\mathbb{F}_q^n\) and \(f: V \to \mathbb{F}_q^n\) is a linear map. Let \(\mathcal{L}\) be the set of all partial linear maps of \(\mathbb{F}_q^n\) containing \(1\). Ordered \(\mathcal{L}\) by ordinary and reverse inclusion, two families of finite posets are obtained. This paper proves that these posets are lattices, discusses their geometricity, and computes their characteristic polynomials.

A total coloring of a 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. An adjacent vertex-distinguishing total coloring \(h\) of a simple graph \(G = (V, E)\) is a proper total coloring of \(G\) such that \(H(u) \neq H(v)\) for any two adjacent vertices \(u\) and \(v\), where \(H(u) = \{h(wu) \mid wu \in E(G)\} \cup \{h(u)\}\) and \(H(v) = \{h(xv) \mid xv \in E(G)\} \cup \{h(v)\}\). The minimum number of colors required for a proper total coloring (resp. an adjacent vertex-distinguishing total coloring) of \(G\) is called the total chromatic number (resp. adjacent vertex-distinguishing total chromatic number) of \(G\) and denoted by \(\chi_t(G)\) (resp. \(\chi_{at}(G)\)). The Total Coloring Conjecture (TCC) states that for every simple graph \(G\), \(\chi(G) + 1 \leq \chi_t(G) \leq \Delta(G) + 2\). \(G\) 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 augmented cube \(AQ_n\) is of Type 1 for \(n \geq 4\). We also consider the adjacent vertex-distinguishing total chromatic number of \(AQ_n\) and prove that \(\chi_{at}(AQ_n) = \Delta(AQ_n) + 2\) for \(n \geq 3 \).

The Channel Assignment Problem is often modeled by integer vertex-labelings of graphs. We will examine \(L(2,1)\)-labelings that realize the span \(\lambda\) of a simple, connected graph \(G = (V, E)\). We define the utility of \(G\) to be the number of possible expansions that can occur on \(G\), where an expansion refers to an opportunity to add a new vertex \(u\) to \(G\), with label \(\lambda(u)\), such that:

1. edges are added between \(u\) and \(v\);
2. edges are added between \(u\) and the neighbors of \(v\); and
3. the resulting labeling of the graph is a valid \(L(2, 1)\)-labeling.

Building upon results of Griggs, Jin, and Yeh, we use known values of \(\lambda\) to compute utility for several infinite families and analyze the utility of specific graphs that are of interest elsewhere.

A Sidon set \(S\) is a set of integers where the number of solutions to any integer equation \(k = k_1 + k_2\) with \(k_1, k_2 \in S\) is at most \(2\). If \(g \geq 2\), the set \(S\) is a generalized Sidon set. We consider Sidon sets modulo \(n\), where the solutions to addition of elements are considered under a given modulus. In this note, we give a construction of a generalized Sidon set modulo \(n\) from any known Sidon set.

In an ordered graph \(G\), a set of vertices \(S\) with a pre-coloring of the vertices of \(S\) is said to be a greedy defining set (GDS) if the greedy coloring of \(G\) with fixed colors of \(S\) yields a \(\chi(G)\)-coloring of \(G\). This concept first appeared in [M. Zaker, Greedy defining sets of graphs, Australas. J. Combin, 2001]. The smallest size of any GDS in a graph \(G\) is called the greedy defining number of \(G\). We show that determining the greedy defining number of bipartite graphs is an NP-complete problem, affirmatively answering a problem mentioned in a previous paper. Additionally, we demonstrate that this number for forests can be determined in linear time. Furthermore, we present a method for obtaining greedy defining sets in Latin squares and, using this method, show that any \(n \times n\) Latin square has a GDS of size at most \(n^2 – (n \log 4n)/4\).

Multi-receiver authentication codes allow one sender to construct an authenticated message for a group of receivers such that each receiver can verify authenticity of the received message. In this paper, we construct one multi-receiver authentication codes from pseudo-symplectic geometry over finite fields. The parameters and the probabilities of deceptions of this codes are also computed.

Resistance distance was introduced by Klein and Randic as a generalization of the classical distance. The Kirchhoff index \(Kf(G)\) of a graph \(G\) is the sum of resistance distances between all pairs of vertices. In this paper, we determine the bicyclic graph of order \(n \geq 8\) with maximal Kirchhoff index. This improves and extends an earlier result by Zhang \(et\; al. [19]\).

Bereg and Wang defined a new class of highly balanced \(d\)-ary trees which they call \(k\)-trees; these trees have the interesting property that the internal path length and thus the Wiener index can be calculated quite easily. A \(k\)-tree is characterized by the property that all levels, except for the last \(k\) levels, are completely filled. Bereg and Wang claim that the number of \(k\)-trees is exponentially increasing, but do not give an asymptotic formula for it. In this paper, we study the number of \(d\)-ary \(k\)-trees and the number of mutually non-isomorphic \(d\)-ary \(k\)-trees, making use of a technique due to Flajolet and Odlyzko.

A group \(G\) is said to be a \(B_k\)-group if for any \(k\)-subset \(\{a_1, \ldots, a_k\}\) of \(G\), \(\left|\{a_ia_j \mid 1 \leq i, j \leq k\}\right| \leq \frac{k(k+1)}{2}\). In this paper, a complete classification of \(B_5\)-groups is given.

For a simple undirected graph \(G = (V, E)\), a subset \(I\) of \(V(G)\) is said to be an independent set of \(G\) if any two vertices in \(I\) are not adjacent in \(G\). A maximal independent set is an independent set that is not a proper subset of any other independent set. In this paper, we survey the largest to fourth largest numbers of maximal independent sets among all trees and forests. In addition, we further look into the problem of determining the fifth largest number of maximal independent sets among all trees and forests. Extremal graphs achieving these values are also given.