We are interested in linear-fractional transformations \(y,t\) satisfying the relations \(y^6=t^6 = 1\), with a view to studying an action of the subgroup \(H = \) on \({Q}(\sqrt{n}) \cup \{\infty\}\) by using coset diagrams.

For a fixed non-square positive integer \(n\), if an element \(\alpha = \frac{a+\sqrt {n}}{c}\) and its algebraic conjugate have different signs, then \(\alpha\) is called an ambiguous number. They play an important role in the study of action of the group \(H\) on \({Q}(\sqrt{n}) \cup \{\infty\}\). In the action of \(H\) on \({Q}(\sqrt{n}) \cup \{\infty\}\), \(\mathrm{Stab}_\alpha{(H)}\) are the only non-trivial stabilizers and in the orbit \(\alpha H\); there is only one (up to isomorphism). We classify all the ambiguous numbers in the orbit and use this information to see whether the action is transitive or not.

We are studying clique graphs of planar graphs, \(K(\text{Planar})\), this means the graphs which are the intersection of the clique family of some planar graph. In this paper, we characterize the \(K_3\) – free and \(K_4\) – free graphs which are in \(K(\text{Planar})\).

We show that a self-complementary vertex-transitive graph of order \(pq\), where \(p\) and \(q\) are distinct primes, is isomorphic to a circulant graph of order \(pq\). We will also show that if \(\Gamma\) is a self-complementary Cayley graph of the nonabelian group \(G\) of order \(pq\), then \(\Gamma\) and the complement of \(\Gamma\) are not isomorphic by a group automorphism of \(G\).

One of the most important problems of coding theory is to construct codes with the best possible minimum distance. The class of quasi-cyclic codes has proved to be a good source for such codes. In this paper, we use the algebraic structure of quasi-cyclic codes and the BCH type bound introduced in [17] to search for quasi-cyclic codes which improve the minimum distances of the best-known linear codes. We construct \(11\) new linear codes over \(\text{GF}(8)\) where \(3\) of these codes are one unit away from being optimal.

A graph \(G\) is said to be \(locally\) \(hamiltonian\) if the subgraph induced by the neighbourhood of every vertex is hamiltonian. Alabdullatif conjectured that every connected locally hamiltonian graph contains a spanning plane triangulation. We disprove the conjecture. At the end, we raise a problem about the nonexistence of spanning planar triangulation in a class of graphs.

Recently, Babson and Steingrimsson (see \([BS]\)) introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation.

In this paper we study the generating functions for the number of permutations on \(n\) letters avoiding a generalized pattern \(ab-c\) where \((a,b,c) \in S_3\), and containing a prescribed number of occurrences of a generalized pattern \(cd-e\) where \((c,d,e) \in S_3\). As a consequence, we derive all the previously known results for this kind of problem, as well as many new results.

Let \(G = (V,E)\) be a simple graph. For any real valued function \(f:V \to {R}\) and \(S \subset V\), let \(f(S) = \sum_{v\in S} f(u)\). A signed \(k\)-subdominating function is a function \(f: V \to \{-1,1\}\) such that \(f(N[v]) \geq 1\) for at least \(k\) vertices \(v \in V\). The signed \(k\)-subdomination number of a graph \(G\) is \(\gamma_{ks}^{-11}(G) = \min \{f(V) | f \text{ is a signed } k\text{-subdominating function on } G\}\). In this paper, we obtain lower bounds on this parameter and extend some results in other papers.

We give some relationships among the intersection numbers of a distance-regular graph \(\Gamma\) which contains a circuit \((u_1,u_2,u_3,u_4)\) with \(\partial(u_1,u_2) = 1\) and \(\partial(u_2,u_4) = 2\). As an application, we obtain an upper bound of the diameter of \(\Gamma\) when \(k \geq 2b_1\).

We extend results concerning orthogonal edge labeling of constant weight Gray codes. For positive integers \(n\) and \(r\) with \(n > r\), let \(G_{n,r}\) be the graph whose vertices are the \(r\)-sets of \(\{1, \ldots, n\}\), with \(r\)-sets adjacent if they intersect in \(r-1\) elements. The graph \(G_{n,r}\) is Hamiltonian; Hamiltonian cycles of \(G_{n,r}\) are early examples of error-correcting codes, where they came to be known as constant weight Gray codes.

An \(r\)-set \(A\) and a partition \(\pi\) of weight \(r\) are said to be orthogonal if every block of \(\pi\) meets \(A\) in exactly one element. Given a class \(P\) of weight \(r\) partitions of \(X_n\), one would like to know if there exists a \(G_{n,r}\) Hamiltonian cycle \(A_1 A_2 \ldots A_{\binom{n}{r}}\) whose edges admit a labeling \(A_1\pi_1 A_2 \ldots A_{\binom{n}{r}}\pi_{\binom{n}{r}}\) by distinct partitions from \(\mathcal{P}\), such that a partition label of an edge is orthogonal to the vertices that comprise the edge. The answer provides non-trivial information about Hamiltonian cycles in \(G_{n,r}\) and has application to questions pertaining to the efficient generation of finite semigroups.

Let \(r\) be a partition of \(m\) as a sum of \(r\) positive integers. We let \(r\) also refer to the set of all partitions of \(X_n\) whose block sizes comprise the partition \(r\). J. Lehel and the first author have conjectured that for \(n \geq 6\) and partition type \(\pi\) of \(\{1, \ldots, n\}\) of weight \(r\) partitions, there exists a \(r\)-labeled Hamiltonian cycle in \(G_{n,r}\).

In the present paper, for \(n = s + r\), we prove that there exist Hamiltonian cycles in \(G_{n,r}\) which admit orthogonal labelings by the partition types which have \(s\) blocks of size two and \(r – s\) blocks of size one, thereby extending a result of J. Lehel and the first author and completing the work on the conjecture for all partition types with blocks of size at most two.

For distinct vertices \(u\) and \(v\) of a nontrivial connected graph \(G\), the detour distance \(D(u,v)\) between \(u\) and \(v\) is the length of a longest \(u-v\) path in \(G\). For a vertex \(v \in V(G)\), define \(D^-(v) = \min\{D(u,v) : u \in V(G) – \{v\}\}\). A vertex \(u (\neq v)\) is called a detour neighbor of \(v\) if \(D(u,v) = D^-(v)\). A vertex \(v\) is said to detour dominate a vertex \(u\) if \(u = v\) or \(u\) is a detour neighbor of \(v\). A set \(S\) of vertices of \(G\) is called a detour dominating set if every vertex of \(G\) is detour dominated by some vertex in \(S\). A detour dominating set of \(G\) of minimum cardinality is a minimum detour dominating set and this cardinality is the detour domination number \(\gamma_D(G)\). We show that if \(G\) is a connected graph of order \(n \geq 3\), then \(\gamma_D(G) \leq n-2\). Moreover, for every pair \(k,n\) of integers with \(1 \leq k \leq n-2\), there exists a connected graph \(G\) of order \(n\) such that \(\gamma_D(G) = k\). It is also shown that for each pair \(a,b\) of positive integers, there is a connected graph \(G\) with domination number \(\gamma(G) = a\) and \(\gamma_D(G) = b\).