A \(G\)-design is a partition of \(E(K_v)\) in which each element induces a copy of \(G\). The existence of \(G\)-designs with the additional property that they contain no proper subsystems has been previously settled when \(G \in \{K_3, K_4 – e\}\). In this paper, the existence of \(P_m\)-designs which contain no proper subsystems is completely settled for every value of \(m\) and \(v\).

The Randić index of an organic molecule whose molecular graph is \(G\) is the sum of the weights \((d(u)d(v))^{-\frac{1}{2}}\) of all edges \(uv\) of \(G\), where \(d(u)\) and \(d(v)\) are the degrees of the vertices \(u\) and \(v\) in \(G\). In this paper, we give a sharp lower bound on the Randić index of cacti with perfect matchings.

Let \(\text{ASG}(2v+1,v;\mathbb{F}_q)\) be the \((2v+1)\)-dimensional affine-singular symplectic space over the finite field \(\mathbb{F}_q\) and let \(\text{ASp}_{2v+1}(\mathbb{F}_q)\) be the affine-singular symplectic group of degree \(2v+1\) over \(\mathcal{F}_q\). For any orbit \(O\) of flats under \(\text{ASp}_{2v+1}(\mathbb{F}_q)\), let \(\mathcal{L}\) be the set of all flats which are intersections of flats in \(O\) such that \(O \subseteq \mathcal{L}\) and assume the intersection of the empty set of flats in \(\text{ASG}(2v+1,v;\mathbb{F}_q)\) is \(\mathbb{F}_q^{2v+1}\). By ordering \(\mathcal{L}\) by ordinary or reverse inclusion, two lattices are obtained. This article discusses the relations between different lattices, classifies their geometricity, and computes their characteristic polynomial.

Let \(\gamma_c(G)\) be the connected domination number of \(G\).A graph is \(k\)-\(\gamma_c\)-critical if \(\gamma_c(G) = k\) and \(\gamma_c(G + uv) < \gamma_c(G)\) for any nonadjacent pair of vertices \(u\) and \(v\) in the graph \(G\). In this paper, we show that the diameter of a \(k\)-\(\gamma_c\)-critical graph is at most \(k\) and this upper bound is sharp.

A \(b\)-coloring of a graph \(G\) by \(k\) colors is a proper \(k\)-coloring of the vertices of \(G\) such that in each color class there exists a vertex having neighbors in all the other \(k-1\) color classes. The \(b\)-chromatic number \(\varphi(G)\) of a graph \(G\) is the maximum \(k\) for which \(G\) has a \(b\)-coloring by \(k\) colors. This concept was introduced by R.W. Irving and D.F. Manlove in \(1999\). In this paper, we study the \(b\)-chromatic numbers of the cartesian products of paths and cycles with complete graphs and the cartesian product of two complete graphs.

Let \(K_{d,d}\) be a complete bipartite digraph. In this paper, we determine the exact value of the domination number in iterated line digraph of \(K_{d,d}\).

A total coloring of a simple graph \(G\) is called adjacent vertex distinguishing if for any two adjacent and distinct vertices \(u\) and \(v\) in \(G\), the set of colors assigned to the vertices and the edges incident to \(u\) differs from the set of colors assigned to the vertices and the edges incident to \(v\). In this paper, we shall prove that the adjacent vertex distinguishing total chromatic number of an outer plane graph with \(\Delta \leq 5\) is \(\Delta+2\) if \(G\) has two adjacent maximum degree vertices, otherwise it is \(\Delta+1\).

Let \(P_j(n)\) denote the number of representations of \(n\) as a sum of \(j\) pentagonal numbers. We obtain formulas for \(P_j(n)\) when \(j = 2\) and \(j = 3\).

Eternal domination of a graph requires the vertices of the graph to be protected, against infinitely long sequences of attacks, by guards located at vertices, with the requirement that the configuration of guards induces a dominating set at all times. We study some variations of this concept in which the configuration of guards induce total dominating sets. We consider two models of the problem: one in which only one guard moves at a time and one in which all guards may move simultaneously. A number of upper and lower bounds are given for the number of guards required.