For each of the parameter sets \((30, 7, 15)\) and \((26, 12, 55)\), a simple \(3\)-design is given. They have \(\text{PSL}(2, 29)\) and \(\text{PSL}(2, 25)\) as their automorphism group, respectively. Each of the two simple \(3\)-designs is the first one ever known with the parameter set given and \(4\) in each of the two parameter sets is minimal for the given \(v\) and \(k\).

In this paper, we study linear codes over finite chain rings. We relate linear cyclic codes, \((1 + \gamma^k)\)-cyclic codes and \((1 – \gamma^k)\)-cyclic codes over a finite chain ring \(R\), where \(\gamma\) is a fixed generator of the unique maximal ideal of the finite chain ring \(R\), and the nilpotency index of \(\gamma\) is \(k+1\). We also characterize the structure of \((1+\gamma^k)\)-cyclic codes and \((1 – \gamma^k)\)-cyclic codes over finite chain rings.

Let \(G\) be a graph with \(n\) vertices. The mean integrity of \(G\) is defined as follows:\(J(G) = min_{P \subseteq V} \{|P| + \tilde{m}(G – P)\},\) where \(\tilde{m}(G – P) = \frac{1}{n-|P|}\sum_{v \in G – P} n_v\) and \(n_v\) is the size of the component containing \(v\). The main result of this article is a formula for the mean integrity of a path \(P_n\) of \(n\) vertices. A corollary of this formula establishes the mean integrity of a cycle \(C_n\) of \(n\) vertices.

It is known that any reducible additive hereditary graph property has infinitely many minimal forbidden graphs, however the proof of this fact is not constructive. The purpose of this paper is to construct infinite families of minimal forbidden graphs for some classes of reducible properties. The well-known Hajós’ construction is generalized and some of its applications are presented.

In this paper, we prove that for any graph \(G\), there is a dominating induced subgraph which is a cograph. Two new domination parameters \(\gamma_{cd}\) – the cographic domination number and \(\gamma_{gcd}\) – the global cographic domination number are defined. Some properties, including complexity aspects, are discussed.

Given a permutation \(\pi\) chosen uniformly from \(S_n\), we explore the joint distribution of \(\pi(1)\) and the number of descents in \(\pi\). We obtain a formula for the number of permutations with \(Des(\pi) = d\) and \(\pi(1) = k\), and use it to show that if \(Des(\pi)\) is fixed at \(d\), then the expected value of \(\pi(1)\) is \(d+1\). We go on to derive generating functions for the joint distribution, show that it is unimodal if viewed correctly, and show that when \(d\) is small the distribution of \(\pi(1)\) among the permutations with \(d\) descents is approximately geometric. Applications to Stein’s method and the Neggers-Stanley problem are presented.

A graph is called induced matching extendable, if every induced matching of it is contained in a perfect matching of it. A graph \(G\) is called \(2k\)-vertex deletable induced matching extendable, if \(G — S\) is induced matching extendable for every \(S \subset V(G)\) with \(|S| = 2k\). The following results are proved in this paper. (1) If \(\kappa(G) \geq \lceil \frac{v(G)}{3} \rceil +1\) and \(\max\{d(u), d(v)\} \geq \frac{2v(G)+1}{3}\) for every two nonadjacent vertices \(u\) and \(v\), then \(G\) is induced matching extendable. (2) If \(\kappa(G) \geq \lceil \frac{v(G)+4k}{3}\rceil\) and \(\max\{d(u), d(v)\} \geq \lceil \frac{2v(G)+2k}{3} \rceil\) for every two nonadjacent vertices \(u\) and \(v\), then \(G\) is \(2k\)-vertex deletable induced matching extendable. (3) If \(d(u) + d(v) \geq 2\lceil\frac{2v(G)+2k}{3} \rceil – 1\) for every two nonadjacent vertices \(u\) and \(v\), then \(G\) is \(2k\)-vertex deletable IM-extendable. Examples are given to show the tightness of all the conditions.