Let \(K\) be a real quadratic field \(\mathbb{Q}(\sqrt{n})\) with an integer \(n = df^2\), where \(d\) is the field discriminant of \(K\) and \(f \geq 1\). Q. Mushtaq found an interesting phenomenon that any totally negative number \(\kappa_0\) with \(\kappa^{\sigma} < 0\) and \(\kappa_0^{\sigma} < 0\) belonging to the discriminant \(n\), attains an ambiguous number \(\kappa_m\) with \(\kappa_m \kappa_m^{\sigma} < 0\) after finitely many actions \(\kappa_0^{A_j}\) with \(0 \leqq j \leqq m\) by modular transformations \(A_j \in \mathrm{SL}_2^+(\mathbb{Z})\). Here \(\sigma\) denotes the embedding of \(K\) distinct from the identity. In this paper, we give a new aspect for the process to reach an ambiguous number from a totally negative or totally positive number, by which the gap of the proof of Q. Mushtaq's Theorem is complemented. Next, as an analogue of Gauss' Genus Theory, we prove that the ring class number \(h_{+}(df^2)\) coincides with the ambiguous class number belonging to the discriminant \(n = df^2\), and its behavior is unbounded when \(f\) with suitable prime factors goes to infinity using the ring class number formula.

For a rational number \(r > 1\), a set \(A\) of positive integers is called an \(r\)-multiple-free set if \(A\) does not contain any solution of the equation \(rx = y\). The extremal problem of estimating the maximum possible size of \(r\)-multiple-free sets contained in \([n] := \{1, 2, \ldots, n\}\) has been studied in combinatorial number theory for theoretical interest and its application to coding theory. Let \(a\) and \(b\) be relatively prime positive integers such that \(a < b\). Wakeham and Wood showed that the maximum size of \((b/a)\)-multiple-free sets contained in \([n]\) is \( \frac{b}{b+1} + O(\log n)\). In this note, we generalize this result as follows. For a real number \(p \in (0, 1)\), let \([n]_p\) be a set of integers obtained by choosing each element \(i \in [n]\) randomly and independently with probability \(p\). We show that the maximum possible size of \((b/a)\)-multiple-free sets contained in \([n]_p\) is \({\frac{b}{b+p}pn} + O(\sqrt{pn} \log n \log \log n)\) with probability that goes to \(1\) as \(n \to \infty\).

A partition of an integer \(n\) is a representation \(n = a_1 + a_2 + \cdots + a_k\), with integer parts \(a_1 \geq a_2 \geq \cdots \geq a_k \geq 1\). The Durfee square is the largest square of points in the graphical representation of a partition. We consider generating functions for the sum of areas of the Durfee squares for various different classes of partitions of \(n\). As a consequence, interesting partition identities are derived. The more general case of Durfee rectangles is also treated, as well as the asymptotic growth of the mean area over all partitions of \(n\).

A graph \(G\) is called a fractional \((k, m)\)-deleted graph if any \(m\) edges are removed from \(G\), then the resulting graph admits a fractional \(k\)-factor. In this paper, we prove that for integers \(k \geq 2\), \(m \geq 0\), \(n \geq 8k + 4m – 7\), and \(\delta(G) \geq k + m\), if
\[|N_G(x) \cup N_G(y)| \geq \frac{n}{2}\]
for each pair of non-adjacent vertices \(x, y\) of \(G\), then \(G\) is a fractional \((k, m)\)-deleted graph. The bounds for neighborhood union condition, order, and the minimum degree of \(G\) are all sharp.

A \(c\)-partite or multipartite tournament is an orientation of a complete \(c\)-partite graph. A digraph \(D\) is cycle complementary if there exist two vertex-disjoint directed cycles \(C\) and \(C’\) such that \(V(D) = V(C) \cup V(C’)\). The global irregularity of a digraph \(D\) is defined by
\[i_g(D) = \max\{\max(d^+(x), d^-(x)) – \min(d^+(y),d^-(y)) \mid x,y \in V(D)\}.\]
If \(i_g(D) = 0\), then \(D\) is regular, and if \(i_g(D) \leq 1\), then \(D\) is almost regular. We prove in this paper that every almost regular \(c\)-partite tournament with \(c \geq 3\) such that all partite sets have the same cardinality \(r \geq 4\) contains two complementary directed cycles of length \(3\) and \(|V(D)| – 3\).

In this paper, we determine the spectrum for \(super-perfect\) OQSs. OQSs are \(G\)-designs in which \(G\) is an octagon quadrangle, i.e., the graph consisting of an \(8\)-cycle \((x_1, x_2, \ldots, x_8)\) with two additional chords: the edges \(\{x_1, x_4\}\) and \(\{x_5, x_6\}\).

In this paper, we give a four parameter theta function identity and prove it by using some properties of Jacobi’s theta functions and Jacobi’s fundamental formulae.

The order dimension is an invariant on partially ordered sets introduced by Dushnik and Miller in \(1941 [1]\). It is known that the computation of the order dimension of a partially ordered set in general is highly complex,with current algorithms relying on the minimal coloring of an associated hypergraph, see \([5]\). The aim of this work is to extend the family of posets whose order dimension is easily determined by a formula. We introduce an operation called layering. Finally, we provide the precise formulas for determining the order dimension of any given number of layers of Trotter’s generalized crowns.

In this paper, the regular endomorphisms of a split graph are investigated. We give a condition under which the regular endomorphisms of a split graph form a monoid.

The clique graph \(K(G)\) of a graph \(G\) is the intersection graph of all its (maximal) cliques, and \(G\) is said to be clique divergent if the order of its \(n\)-th iterated clique graph \(K^n(G)\) tends to infinity with \(n\). In general, deciding whether a graph is clique divergent is not known to be computable. We characterize the dynamical behavior under the clique operator of circulant graphs of the form \(C_n(a, b, c)\) with \(0 < a < b < c < \frac{n}{3}\). Such a circulant is clique divergent if and only if it is not clique-Helly. Owing to the Dragan-Szwarcfiter Criterion to decide clique-Hellyness, our result implies that the clique divergence of these circulants can be decided in polynomial time. Our main difficulty was the case \(C_n(1, 2, 4)\), which is clique divergent but no previously known technique could be used to prove it.