The study of the generalized Fermat variety

\[
\phi_j = \frac{x^j + y^j + z^j + (x+y+z)^j}{(x+y)(x+z)(y+z)}
\]

defined over a finite field \( L = \mathbb{F}_q \), where \( q = 2^n \) for some positive integer \( n \), plays an important role in the study of (APN) functions and exceptional APN functions. This study arose after a characterization by Rodier that relates these functions with the number of rational points of \( \phi_j = (x,y,z) \). The most studied cases are when \( j = 2^k + 1 \) and \( j = 2^{2k} – 2^{k} + 1 \), the Gold and Kasami-Welch numbers. In this article, we make a claim about the decomposition of \( \phi_j \) into absolutely irreducible components. If these components intersect transversally at a particular point, then the corresponding Kasami-Welch polynomial is absolutely irreducible. This implies that the function is not exceptional APN, thus helping us make progress on the stated conjecture.

For \( n \geq 1 \), let \( a_n \) count the number of strings \( s_1 s_2 s_3 \ldots s_n \), where
(i) \( s_1 = 0 \);
(ii) \( s_i \in \{0, 1, 2\} \) for \( 2 \leq i \leq n \);
(iii) \( |s_i – s_{i-1}| \leq 1 \) for \( 2 \leq i \leq n \).

Then \( a_1 = 1 \), \( a_2 = 2 \), \( a_3 = 5 \), \( a_4 = 12 \), and \( a_5 = 29 \).

In general, for \( n \geq 3 \), \( a_n = 2a_{n-1} + a_{n-2} \), and \( a_n \) equals \( P_n \), the \( n \)th \emph{Pell} number.

For these \( P_n \) strings of length \( n \), we count
(i) The number of occurrences of each symbol \( 0, 1, 2 \);
(ii) The number of times each symbol \( 0, 1, 2 \) occurs in an even or odd position;
(iii) The number of levels, rises, and descents within the strings;
(iv) The number of runs that occur within the strings;
(v) The sum of all strings considered as base \( 3 \) integers;
(vi) The number of inversions and coinversions within the strings; and
(vii) The sum of the major indices for the strings.

A family of graphs, called Generalized Johnson graphs, provides an abstraction of both Kneser and Johnson graphs.
Given the symmetric nature of Generalized Johnson graphs, we provide various decompositions of these graphs and demonstrate non-trivial instances of the impossibility of decomposing such graphs into triples.

A branch vertex of a tree is a vertex of degree at least three. Matsuda, et. al. [7] conjectured that, if \(n\) and \(k\) are non-negative integers and \(G\) is a connected claw-free graph of order \(n\), there is either an independent set on \(2k + 3\) vertices whose degrees add up to at most \(n – 3\), or a spanning tree with at most \(k\) branch vertices. The authors of the conjecture proved it for \(k = 1\); we prove it for \(k = 2\).

Orbit code is a class of constant dimension code which is defined as orbit of a subgroup of the general linear group \(GL_n(\mathbb{F}q)\), acting on the set of all the subspaces of vector space \(\mathbb{F}_q^n\). In this paper, the construction of orbit codes based on the singular general linear group \(GL{n+l, n}(\mathbb{F}_q)\) acting on the set of all the subspaces of type \((m, k)\) in singular linear spaces \(\mathbb{F}_q^{n+l}\) is given. We give a characterization of orbit code constructed in singular linear space \(\mathbb{F}_q^{n+l}\), and then give some basic properties of the constructed orbit codes. Finally, two examples about our orbit codes for understanding these properties explicitly are presented.

We use heuristic algorithms to find terraces for small groups. We show that Bailey’s Conjecture (that all groups other than the non-cyclic elementary abelian 2-groups are terraced) holds up to order 511, except possibly at orders 256 and 384. We also show that Keedwell’s Conjecture (that all non-abelian groups of order at least 10 are sequenceable) holds up to order 255, and for the groups \(A_6, S_6, PSL(2, q_1)\) and \(PGL(2, q_2)\) where \(q_1\) and \(q_2\) are prime powers with \(3 \leq q_1 \leq 11\) and \(3 \leq q_2 \leq 8\). A sequencing for a group of a given order implies the existence of a complete Latin square at that order. We show that there is a sequenceable group for each odd order up to 555 at which there is a non-abelian group. This gives 31 new orders at which complete Latin squares are now known to exist, the smallest of which is 63.

In addition, we consider terraces with some special properties, including constructing a directed \(T_2\)-terrace for the non-abelian group of order 21 and hence a Roman-2 square of order 21 (the first known such square of odd order). Finally, we report the total number of terraces and directed terraces for groups of order at most 15.

An adjacent vertex distinguishing total coloring of a graph \(G\) is a proper total coloring of \(G\) such that no pair of adjacent vertices are incident to the same set of colors. The minimum number of colors required for an adjacent vertex distinguishing total coloring of \(G\) is denoted by \(\chi”_a(G)\). In this paper, we prove that if \(G\) is an outer 1-planar graph with at least two vertices, then \(\chi”_a(G) \leq \max\{\Delta + 2, 8\}\). Moreover, we also prove that when \(\Delta \geq 7\), \(\chi”_a(G) = \Delta + 2\) if and only if \(G\) contains two adjacent vertices of maximum degree.

In this paper, we study five methods to construct \(\alpha\)-trees by using vertex amalgamations of smaller \(\alpha\)-trees. We also study graceful and \(\alpha\)-labelings for graphs that are the union of \(t\) copies of an \(\alpha\)-graph \(G\) of order \(m\) and size \(n\) with a graph \(H\) of size \(t\). If \(n > m\), we prove that the disjoint union of \(H\) and \(t\) copies of \(G\) is graceful when \(H\) is graceful, and that this union is an \(\alpha\)-graph when \(H\) is any linear forest of size \(t – 1\). If \(n = m\), we prove that this union is an \(\alpha\)-graph when \(H\) is the path \(P_{t-1}\).

For a bipartite graph \( G \) and a positive integer \( s \), the \( s \)-bipartite Ramsey number \( BR_s(G) \) of \( G \) is the minimum integer \( t \) with \( t \geq s \) for which every red-blue coloring of \( K_{s,t} \) results in a monochromatic \( G \). A formula is established for \( BR_s(K_{r,r}) \) when \( s \in \{2r – 1, 2r\} \) when \( r \geq 2 \). The \( s \)-bipartite Ramsey numbers are studied for \( K_{3,3} \) and various values of \( s \). In particular, it is shown that \( BR_5(K_{3,3}) = 41 \) when \( s \in \{5,6\} \), \( BR_s(K_{3,3}) = 29 \) when \( s \in \{7,8\} \), and \( 17 \leq BR_{10}(K_{3,3}) \leq 23 \).