We point out that restricted SB triple systems can only exist for \(v \leq 8\). The case \(v = 8\) is especially interesting since it is extremal in that the pair frequencies of the fifteen pairs not involving either \(1\) or \(2\) must be the frequencies \(2, 3, \dots, 16\), in some order.

Let \( a \) and \( b \) be two positive integers. For the graph \( G \) with vertex set \( V(G) \) and edge set \( E(G) \) with \( p = |V(G)| \) and \( q = |E(G)| \), we define two sets \( Q(a) \) and \( P(b) \) as follows:

\[
Q(a) = \begin{cases}
\{\pm a, \pm(a+1), \ldots, \pm(a + (q-2)/2)\} & \text{if } q \text{ is even,} \\
\{0\} \cup \{\pm a, \pm(a+1), \ldots, \pm(a + (q-3)/2)\} & \text{if } q \text{ is odd,}
\end{cases}
\]

\[
P(b) = \begin{cases}
\{\pm b, \pm(b+1), \ldots, \pm(b + (p-2)/2)\} & \text{if } p \text{ is even,} \\
\{0\} \cup \{\pm b, \pm(b+1), \ldots, \pm(b + (p-3)/2)\} & \text{if } p \text{ is odd.}
\end{cases}
\]

For the graph \( G \) with \( p = |V(G)| \) and \( q = |E(G)| \), \( G \) is said to be \( Q(a)P(b) \)-super edge-graceful (in short, \( Q(a)P(b) \)-SEG), if there exists a function pair \( (f, f^+) \) which assigns integer labels to the vertices and edges; that is, \( f^+: V(G) \to P(b) \), and \( f: E(G) \to Q(a) \) such that \( f^+ \) is onto \( P(b) \) and \( f \) is onto \( Q(a) \), and

\[
f^+(u) = \sum\{ f(u,v) : (u, v) \in E(G) \}.
\]

We investigate \( Q(a)P(b) \) super-edge-graceful labelings for three classes of \( (p,p+1) \)-graphs.

The Ramsey number \( R(C_p, C_q, C_r) \) is the smallest positive integer \( m \) such that no matter how one colors the edges of the \( K_m \) in red, white, and blue, there must be a red \( C_p \), a white \( C_q \), or a blue \( C_r \). In this work, we verified some known \( R(C_p, C_q, C_r) \) values and computed some new \( R(C_p, C_q, C_r) \) values. The results are based on computer algorithms.

A \( (p,q) \) graph \( G \) is total edge-magic if there exists a bijection \( f: V \cup E \to \{1, 2, \ldots, p+q\} \) such that for each \( e = (u,v) \in E \), we have \( f(u) + f(e) + f(v) \) as a constant. For a graph \( G \), denote \( M(G) \) the set of all total edge-magic labelings. The magic strength of \( G \) is the minimum of all constants among all labelings in \( M(G) \), denoted by \( \text{emt}(G) \). The maximum of all constants among \( M(G) \) is called the maximum magic strength of \( G \) and denoted by \( \text{eMt}(G) \).

Hegde and Shetty classify a magic graph as strong if \( \text{emt}(G) = \text{eMt}(G) \), ideal magic if \( 1 \leq \text{eMt}(G) – \text{emt}(G) \leq p \), and \textbf{weak magic} if \( \text{eMt}(G) – \text{emt}(G) > p \). A total edge-magic graph is called a super edge-magic if \( f(V(G)) = \{1, 2, \ldots, p\} \). The problem of identifying which kinds of super edge-magic graphs are weak-magic graphs is addressed in this paper.

For even codeword length \( n = 2k, k > 1 \) and alphabet size \( \sigma > 1 \), a family of comma-free codes is constructed with

\[
{\left\lfloor \frac{\sigma^2}{3} \right\rfloor}^r \left( \sigma^2 – \left\lfloor \frac{\sigma^2}{3} \right\rfloor \right)^{k-r}
\]

codewords where \( 1 \leq r < k \). In particular, a new maximal comma-free code with \( n = 4 \) and \( \sigma = 4 \) is given by one of these codes.

If \( K \) is an \( r \)-clique of \( G \) and \( \chi(G) \) decreases by \( r \) upon the removal of all of the vertices in \( K \), then \( K \) is called a critical \( r \)-clique. Two critical cliques are completely independent provided that no vertex in one clique is adjacent to a vertex from the other. An infinite family of graphs is constructed which demonstrates that for every \( s, t \in \mathbb{N} \), there exists a vertex critical graph which admits a critical \( s \)-clique and a critical \( t \)-clique that are completely independent.

In this paper, we obtain a set of inequalities which are necessary conditions for the existence of balanced arrays of strength five, having \( m \) rows (constraints), and with two symbols. We discuss the use of these inequalities to obtain an upper bound on \( m \), and present some illustrative examples.

For a graph \( G \) with vertex set \( V(G) \) and edge set \( E(G) \), let \( i(G) \) be the number of isolated vertices in \( G \). The \emph{isolated toughness} of \( G \) is defined as

\[
I(G) = \min\left\{\frac{|S|}{i(G-S)} \mid S \subseteq V(G), i(G-S) \geq 2 \right\},
\]

if \( G \) is not complete; and \( I(K_n) = n-1 \). In this paper, we investigate the existence of \([a, b]\)-factors in terms of this graph invariant. We proved that if \( G \) is a graph with \( \delta(G) \geq a \) and \( I(G) \geq a \), then \( G \) has a fractional \( a \)-factor. Moreover, if \( \delta(G) \geq a \), \( I(G) > (a-1) + \frac{a-1}{b} \), and \( G-S \) has no \( (a-1) \)-regular component for any subset \( S \) of \( V(G) \), then \( G \) has an \([a, b]\)-factor. The latter result is a generalization of Katerinis’ well-known theorem about \([a, b]\)-factors (P. Katerinis, Toughness of graphs and the existence of factors, \emph{Discrete Math}. 80(1990), 81-92).

We partition the set of spanning trees contained in the complete graph \( K_n \) into spanning trees contained in the complete bipartite graph \( K_{s,t} \). This classification shows that some properties of spanning trees in \( K_n \) can be derived from trees in \( K_{s,t} \). We use Abel’s binomial theorem and the formula for spanning trees in \( K_{s,t} \) to obtain a proof of Cayley’s theorem using partial derivatives. Some results concerning non-isomorphic spanning trees are presented. In particular, we count these trees for \( Q_3 \) and the Petersen graph.

We introduce the ring of ordinomials, which will be utilized in defining the partial chromatic ordinomials of infinite graphs with certain properties – a generalization of chromatic polynomials of finite graphs.

In algebraic contexts, Weyl group elements are usually represented in terms of generators and relations, where representation is not unique. For computational purposes, a more combinatorial representation for elements of classical Weyl groups as signed permutation vectors was introduced in [5]. This paper characterizes some special classes of Weyl group elements using this notation. These classes are especially useful for the study of symmetric spaces and their representations.

Let \( G \) be a graph with vertex set \( V(G) \) and edge set \( E(G) \). A labeling \( f: V(G) \to \mathbb{Z}_2 \) induces an edge labeling \( f^*: E(G) \to \mathbb{Z}_2 \), defined by \( f^*(xy) = f(x) + f(y) \), for each edge \( xy \in E(G) \). For \( i \in \mathbb{Z}_2 \), let

\[
\text{v}_f(i) = \text{card}\{ v \in V(G) : f(v) = i \}
\]

and

\[
\text{e}_f(i) = \text{card}\{ e \in E(G) : f^*(e) = i \}.
\]

A labeling \( f \) of a graph \( G \) is said to be friendly if

\[
\lvert \text{v}_f(0) – \text{v}_f(1) \rvert \leq 1.
\]

The friendly index set of the graph \( G \), \( FI(G) \), is defined as

\[
\{ \lvert \text{e}_f(0) – \text{e}_f(1) \rvert : \text{the vertex labeling } f \text{ is friendly} \}.
\]

This is a generalization of graph cordiality. We introduce a graph construction called the root-union and investigate when gaps exist in the friendly index sets of root-unions of stars by cycles.

Proposed in 1942, the Graph Reconstruction Conjecture posits that every simple, finite, undirected graph with three or more vertices can be reconstructed up to isomorphism to the original graph, given the multiset of subgraphs produced by deleting each vertex along with its incident edges. Related to this Reconstruction Conjecture, existential reconstruction numbers, \( \exists rn(G) \), concern the minimum number of vertex-deleted subgraphs required to identify a graph up to isomorphism.

We discuss the resulting data from calculating reconstruction numbers for all simple, undirected graphs with up to ten vertices. From this data, we establish the reasons behind all high existential reconstruction numbers (\( \exists rn(G) > 3 \)) for \( |V(G)| \leq 10 \) and identify new classes of graphs that have high reconstruction numbers for \( |V(G)| > 10 \).

We also consider 2-reconstructibility—the ability to reconstruct a graph \( G \) from the multiset of subgraphs produced by deleting each combination of two vertices from \( G \). The 2-reconstructibility of all graphs with nine or fewer vertices was tested, identifying four graphs in this range with five vertices as the highest order of graphs that are not 2-reconstructible.

A weighing matrix \( W(n, k) \) of order \( n \) with weight \( k \) is an \( n \times n \) matrix with entries from \( \{0, 1, -1\} \) which satisfies \( WW^T = kI_n \). Such a matrix is group-developed if its rows and columns can be indexed by elements of a finite group \( G \) so that \( w_{g,h} = w_{gf,hf} \) for all \( g,h,f \) in \( G \). Group-developed weighing matrices are a natural generalization of perfect ternary arrays and Hadamard matrices. They are closely related to difference sets.

We describe a search for weighing matrices with order 60 and weight 25, developed over solvable groups. There is one known example of a \( W(60, 25) \) developed over a non-solvable group; no solvable examples are known.

We use techniques from representation theory, including a new viewpoint on complementary quotient images, to restrict solvable examples. We describe a computer search strategy which has eliminated two of twelve possible cases. We summarize plans to complete the search.

A \( (p,q) \)-graph \( G \) is said to be edge graceful if the edges can be labeled by \( 1, 2, \ldots, q \) so that the vertex sums are distinct, mod \( p \). It is shown that if a tree \( T \) is edge-graceful, then its order must be odd. Lee conjectured that all trees of odd orders are edge-graceful. In [7], we establish that every tree of odd order with one even vertex is edge-graceful. Mitchem and Simoson [19] introduced the concept of super edge-graceful graphs, which is a stronger concept than edge-graceful for trees. We show that every tree of odd order with three even vertices is super edge-graceful.

It is known that there is not any non-trivial graph with vertices of distinct degrees, and any non-trivial graph must have at least two vertices of the same degree. In this article, we will consider the concept of \( P_3 \)-degree of vertices and will introduce a class of connected graphs with exactly two vertices of the same \( P_3 \)-degree. Also, the graphs with distinct \( P_3 \)-degree vertices will be constructed and it will be proven that for any \( n \geq 6 \) there is at least one graph of order \( n \), with distinct \( P_3 \)-degree vertices.

Let \( a, b \) be two positive integers. A \( (p, q) \)-graph \( G \) is said to be \( Q(a)P(b) \)-super edge-graceful, or simply \( (a, b) \)-SEG, if there exist onto mappings \( f : E(G) \to Q(a) \) and \( f^* : V(G) \to P(b) \), where

\[
Q(a) = \begin{cases}
\{\pm a, \pm(a+1), \ldots, \pm(a + (q-2)/2)\} & \text{if } q \text{ is even}, \\
\{0, \pm a, \pm(a+1), \ldots, \pm(a + (q-3)/2)\} & \text{if } q \text{ is odd},
\end{cases}
\]

\[
P(b) = \begin{cases}
\{\pm b, \pm(b+1), \ldots, \pm(b + (p-2)/2)\} & \text{if } p \text{ is even}, \\
\{0, \pm b, \pm(b+1), \ldots, \pm(b + (p-3)/2)\} & \text{if } p \text{ is odd},
\end{cases}
\]

such that \( f^*(v) = \sum_{uv \in E(G)} f(uv) \). We find the values of \( a \) and \( b \) for which the hypercube \( Q_n, n \leq 3 \), is \( (a, b) \)-SEG.

A map is a graph that admits an orientation of its edges so that each vertex has out-degree exactly \(1\). We characterize graphs which admit a decomposition into \(k\) edge-disjoint maps after: (1) the addition of any \(\ell\) edges; (2) the addition of some \(\ell\) edges. These graphs are identified with classes of \emph{sparse} graphs; the results are also given in matroidal terms.

A connected graph on three or more vertices is said to be irreducible if it has no leaves, and if each vertex has a unique neighbor set. A connected graph on one or two vertices is also said to be irreducible, and a disconnected graph is irreducible if each of its connected components is irreducible. In this paper, we study the class of irreducible graphs. In particular, we consider an algorithm that, for each connected graph \( \Gamma \), yields an irreducible subgraph \( I(\Gamma) \) of \( \Gamma \). We show that this subgraph is unique up to isomorphism. We also show that almost all graphs are irreducible. We then conclude by highlighting some structural similarities between \( I(\Gamma) \) and \( \Gamma \).

A Latin square of order \( n \) is an \( n \) by \( n \) array in which every row and column is a permutation of a set \( N \) of \( n \) elements. Let \( L = [l_{i,j}] \) and \( M = [m_{i,j}] \) be two Latin squares of even order \( n \), based on the same \( N \)-set. Define the superposition of \( L \) onto \( M \) to be the \( n \) by \( n \) array \( A = (l_{i,j}, m_{i,j}) \). When \( n \) is even, \( L \) and \( M \) are said to be \emph{nearly orthogonal} if the superposition of \( L \) onto \( M \) has every ordered pair \( (i, j) \) appearing exactly once except for \( i = j \), when the ordered pair appears \( 0 \) times and except for \( i – j = \frac{n}{2} \pmod{n} \), when the ordered pair appears \( 2 \) times. A set of \( t \) Latin squares of order \( 2m \) is called a set of \emph{mutually nearly orthogonal Latin squares} (MNOLS(\(2m\))) if the \( t \) Latin squares are pairwise nearly orthogonal. We provide two elementary proofs for results that were stated and proved earlier. We also provide some computer results and prove two recursive constructions for MNOLS. Using these results we show that there always exist \( 3 \) mutually nearly orthogonal Latin squares of order \( 2m \), for \( 2m \geq 358 \).