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.

Bicyclic graphs are connected graphs in which the number of edges equals the number of vertices plus one. In this paper, we determine the first three graphs among all bicyclic graphs with \(n\) vertices, ordered according to their least eigenvalues in increasing order.

The modified Zagreb indices are topological indices which reflect certain structural features of organic molecules. In this paper we study the modified Zagreb indices of joins and compositions.

In \([1]\), well-ordered Steiner triple systems were introduced and used to construct \(1\)-perfect partitions of the \(n\)-cube. However, non-trivial well-ordered Steiner triple systems were only known to exist when \(v =15\). In this short note, we present a simple construction to give a non-trivial well-ordered Steiner triple system of order \(v = 2^n – 1\) for all \(n \geq 5\) and this settles a problem in \([1]\).

Different neighbor conditions are considered in \([3,4,9]\) for a graph up-embeddable. In this paper, we consider the neighbor conditions of all the pairs of vertices with diameter \(2\) and obtain the following new result: if \(|N_G(u) \cap N_G(v)| \geq 2\) for any two vertices \(u,v \in D\) where \(D = \{(u, v) | d_G(u, v) = 2, u,v \in V(G)\}\), then \(G\) is up-embeddable.

We study the factorisations of a cyclic permutation of length \(n\) as a product of a minimal number of transpositions, calculating the number \(f(n, m)\) of factorisations in which a fixed element is moved \(m\) times. In this way, we also give a new proof-in the spirit of Clarke’s proof of Cayley’s theorem on the number of labelled trees-of the fact that there are \(n^{n-2}\) such factorisations.

We show that there are relationships between a generalized Lucas sequence and the permanent and determinant of some Hessenberg matrices.

Suppose \(G\) is a simple graph with average vertex degree greater than \(k – 2\). Erdős and Sós conjectured that \(G\) contains every tree on \(k\) vertices. Sidorenko proved \(G\) contains every tree that has a vertex \(v\) with at least \(\left\lfloor\frac{k}{2}\right\rfloor – 1\) leaf neighbors. We prove this is true if \(v\) has only \(\left\lceil\frac{k}{2}\right\rceil – 2\) leaf neighbors. We generalize Sidorenko’s result by proving that if \(G\) has minimum degree \(d\), then \(G\) contains every tree that has a vertex with at least \((k – 1) – d\) leaf neighbors. We use these results to prove that if \(G\) has average degree greater than \(k – 2\) and minimum degree at least \(k – 4\), then \(G\) contains every tree on \(k\) vertices.