For any positive integer $\mathrm{n}$, let ${\mathrm{M}}_{\mathrm{p}}$ contain the prime numbers less than $\mathrm{n}$. Assuming ${\mathrm{M}}_{\mathrm{p}}$ as the set of moduli, we draw a graph with the vertex set $\{1,2,3,\cdots ,\mathrm{n}\}$, and an edge will be built between the vertices $\mathrm{p}$ and $\mathrm{q}$ if and only if $\mathrm{p}\equiv (\mathrm{q}mod\mathrm{m})$ for some $\mathrm{m}\in$ ${\mathrm{M}}_{\mathrm{p}}.$ We call this graph a prime congruence simple graph and label this graph as $\mathrm{G}(\mathrm{n},{\mathrm{M}}_p)$. The objective of this work is to characterize prime congruence simple graphs, and afterwards, by utilizing these graphs, solutions to the system of linear congruences are suggested and demonstrated by applying modular arithmetic. It is shown that this graph is always a connected graph. The generalized formulae for vertex degrees, size, chromatic number, domination number, clique number, and eccentricity of the prime congruence simple graphs are proposed and proved. Also, independence numbers as well as a covering number for the proposed graph through vertices and edges are evaluated. Lastly, as an application of prime congruence simple graphs, the solution of a system of linear congruences is discussed in terms of the degrees of the vertices.

A connected acyclic graph in which the degree of every vertex is at most four is called a molecular tree. A number associated with a molecular tree that can help to approximate the physical or chemical properties of the corresponding molecule is called a topological index. It is of great importance to investigate the relation between the structure and the thermodynamic properties of those molecules. In this paper, we investigated the extreme value of the first reformulated Zagreb index with a given order and degree of a graph. Further, we investigated the molecular trees that attain the maximum and minimum values. As an application, we presented the regression models to predict the acentric factor and entropy of octane isomers. Our extremal graphs give the minimum and the maximum acentric factor and entropy, which satisfied the experimental values.

Using the binary operation " $\Lsh$" on a GE-algebra $X$ given by $\Lsh (x,y)=(y\ast x)\ast x$ and the GE-endomorphism $\mathrm{\Omega }:X\to X$, the notion of ${\mathrm{\Omega }}_{(l,r)}$-endomorphic (resp., ${\mathrm{\Omega }}_{(r,l)}$-endomorphic) GE-derivation is introduced, and several properties are investigated. Also, examples that illustrate these are provided. Conditions under which ${\mathrm{\Omega }}_{(l,r)}$-endomorphic GE-derivations or ${\mathrm{\Omega }}_{(l,r)}$-endomorphic GE-derivations to satisfy certain equalities and inequalities are studied. We explored the conditions under which $f$ becomes order preserving when $f$ is an ${\mathrm{\Omega }}_{(l,r)}$-endomorphic GE-derivation or an ${\mathrm{\Omega }}_{(r,l)}$-endomorphic GE-derivation on $X$. The $f$-kernel and $\mathrm{\Omega }$-kernel of $f$ formed by the ${\mathrm{\Omega }}_{(r,l)}$-endomorphic GE-derivation or ${\mathrm{\Omega }}_{(l,r)}$-endomorphic GE-derivation turns out to be GE-subalgebras. It is observed that the $\mathrm{\Omega }$-kernel of $f$ is a GE-filter of $X$. The condition under which the $f$-kernel of $f$ formed by the ${\mathrm{\Omega }}_{(r,l)}$-endomorphic GE-derivation or ${\mathrm{\Omega }}_{(l,r)}$-endomorphic GE-derivation becomes a GE-filter is explored.