This research targets the investigation of characteristics within the Maximal product of two RL-graphs by scrutinizing particular types of RL-graphs. Our first step in this quest entails introducing RL-graph concepts, followed by defining what constitutes a strong RL-graph, further elucidated by a practical example. Subsequently, we lay out the connection between RL-graphs and their Maximal products. In particular, a theorem establishes that two RL-graphs are regular if their Maximal product maintains regularity, and a parallel rule applies to α-regular RL-graphs. Contrarily, the reverse is not inherently true, a claim supported by a specific example. Nonetheless, by incorporating an additional condition, we validate the converse. Lastly, we assert that two RL-graphs are connected only if their Maximal product is also a connected RL-graph. In conclusion, the Maximal product of two RL-graphs holds potential in modeling societal health metrics and road accident rates.

A $k$-CEC graph is a graph $G$ which has connected domination number $\gamma_{c}(G) = k$ and $\gamma_{c}(G + uv) < k$ for every $uv \in E(\overline{G})$. A $k$-CVC graph $G$ is a $2$-connected graph with  $\gamma_{c}(G) = k$ and $\gamma_{c}(G - v) < k$ for any $v \in V(G)$. A graph is said to be Maximal $k$-CVC if it is both $k$-CEC and $k$-CVC. Let $\delta$, $\kappa$, and $\alpha$ be the minimum degree, connectivity, and independence number of $G$, respectively. In this work, we prove that for a Maximal $3$-CVC graph, if $\alpha = \kappa$, then $\kappa = \delta$. We additionally consider the class of Maximal $3$-CVC graphs with $\alpha < \kappa$ and $\kappa < \delta$, and prove that every $3$-connected Maximal $3$-CVC graph when $\kappa < \delta$ is Hamiltonian connected.

Let G = (V (G), E(G)) be a graph with the set of vertices V (G) and the set of edges E(G). A subset S of E(G) is called a k-nearly independent edge subset if there are exactly k pairs of elements of S that share a common end. Zk(G) is the number of such subsets. This paper studies Z1. Various properties of Z1 are discussed. We characterize the two n-vertex trees with the smallest Z1, as well as the one with the largest value. A conjecture on the n-vertex tree with the second-largest Z1 is proposed.

We investigated Maximal Prym varieties on  nite  elds by attaining their upper bounds on the number of rational points. This concept gave us a motivation for de ning a generalized de ni-tion of Maximal curves i. e., Maximal morphisms. By MAGMA, we give some non-trivial examples of Maximal morphisms that results in non-trivial examples of Maximal Prym varieties.

We determine the forbidden induced subgraphs for the intersection of the classes of chordal bipartite graphs and line graphs of acyclic directed graphs. This is a first step towards finding the forbidden induced subgraphs for the class of line graphs of directed graphs.