In this paper, we present a numerical solution to the fractional differential equation governing water pollution. The Caputo derivative will be used to model water pollution. For the numerical solution, we propose a fractional numerical scheme and use it to generate figures with mathematical software. We analyze qualitative properties of the determination of equilibrium points and examine their stability. We have analyzed the influence of the order of the fractional derivative on modeling pollution problems.

Following [1], we provide a technical paper. This paper addresses the minimum dominating set problem and generalizes the results for graphs with maximum degree 3 to general graphs. In addition, it complements the technical aspects of the results.

The diminished Sombor index \(DSO(G)\) of a graph \(G\) is defined as \[DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},\] where \(d_u\) denotes the degree of vertex \(u\). Recently, this index has attracted considerable attention due to its promising chemical and mathematical properties. While the graphs minimizing the \(DSO\) index among various graph classes have been characterized, little is known about the next smallest values. In this paper, we extend these investigations by determining the extremal graphs that achieve the second through sixth minimum \(DSO\) indices. Specifically, for \(n\)-vertex trees, we identify all trees attaining the second to sixth smallest \(DSO\) values; for \(n\)-vertex unicyclic graphs, we characterize those with the second to sixth smallest \(DSO\) indices; and for \(n\)-vertex bicyclic graphs, we determine the graphs corresponding to the second to fifth smallest \(DSO\) indices. Our results provide a finer ordering of graphs by the diminished Sombor index and contribute to the systematic understanding of its extremal behavior.

Let \(\zeta(G)\) denote the number of minimum dominating sets of a graph \(G\). We determine how local structural constraints affect the multiplicity of optimal domination in several tree families. For path-based pendant constructions, a sharp threshold separates independent choice from complete forcing: attaching one pendant to each path vertex gives \(\zeta(G)=2^{\gamma(G)}\), while attaching at least two pendants at each vertex forces a unique minimum dominating set. Intermediate attachment patterns give constrained growth: removing the endpoint pendants gives \(\zeta(G)=2^{\gamma(G)-2}\), and alternating attachments give Fibonacci behavior \(\zeta(G)\asymp\varphi^{\gamma(G)}\), where \(\varphi=(1+\sqrt5)/2\). These path-based cases realize the spectral bases \(2\), \(\varphi\), and \(1\) through explicit linear recurrences. For complete binary trees \(T_h\), we prove the period-\(3\) law \(\zeta(T_h)\in\{1,3\}\), depending only on \(h\bmod 3\). We also prove that deleting a single leaf preserves \(\gamma\) and doubles \(\zeta\). More generally, if \(X\subseteq L_h\) is sparse, in the sense that no parent in \(L_{h-1}\) loses both leaf children, then \(\zeta(T_h-X)\le 2^{m_1(X)}\zeta(T_h)\), where \(m_1(X)\) counts the parents in \(L_{h-1}\) that lose exactly one child.

Let \(G=(V,E)\) be a simple connected graph. For a vertex \(x\in V(G)\), its degree is denoted by \(d_G(x)\). The diminished Sombor index of \(G\) is defined as
\[
\mathrm{DSO}(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_G(u)^2+d_G(v)^2}}{d_G(u)+d_G(v)}.
\]
In this paper we establish a sharp upper bound on the diminished Sombor index in terms of the chromatic number and a sharp lower bound in terms of the girth of a graph. For each bound, we characterize the graphs that attain equality.

This paper investigates the existence of pair mean cordial (PMC) labelings for several classes of graphs, namely, \(m\) copies of paths, \(m\) copies of cycles, spider graphs, generalized theta graphs, Tunjung graphs, and volcano graphs.

Consider a near-ring \(\Omega\) with a semigroup ideal \(\mathscr{B}\) and a Jordan ideal \(\mathscr{K}\). We analyze the structural properties of prime near-rings through a nonzero derivation \(\Lambda:\Omega \to \Omega\) by imposing suitable conditions to specific subsets. The primary objective is to establish results that lead to significant structural constraints and ensuring the existence of a nonzero derivation, induces commutativity of \(\Omega\). Our results extend prior findings in the literature by providing new insights into the structural properties of near-rings. Additionally, we include an illustrative example to validate our theoretical claims.

In this article, we investigate the boundedness and qualitative behavior of a new class of generalized Hardy-Hilbert iterated integral operators. By introducing a non-separable kernel characterized by a continuous, strictly convex homogeneous denominator \(\Phi(t,\tau)\) of degree \(\gamma > 1\) and a rational boundary profile, we establish rigorous two-sided bounds within weighted Lebesgue spaces. While typical approaches in the literature focus on one-sided inequalities under rigid symmetry assumptions, our analytical framework addresses mass dissipation near singular boundaries by restricting the operator core to a compact truncated conical sector \(\Omega_c\). We prove an upper bound criterion under explicit weight compatibility conditions and derive a non-trivial positive lower bound proportional to the local \(L_p\)-norm of the input function restricted to the active domain of \(\Omega_c\). Furthermore, we provide illustrative examples and correct previous formulations regarding metric transformations.