Open Journal of Discrete Applied Mathematics (ODAM)

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.

Graphs describe pairwise relations, while hypergraphs represent interactions involving more than two vertices. Super-HyperGraphs further permit vertices to be selected from iterated powersets, so that incidences can occur among nested objects such as teams, clusters, departments, portfolios, or control units. Many systems with such hierarchical organization are also time-dependent: their relations appear, disappear, or change activity over discrete or continuous time. Existing temporal graphs and temporal hypergraphs record temporal activation of edges or hyperedges, but they usually operate over a single-level vertex domain and therefore do not retain the identity of higher-level interacting objects. This paper develops the temporal \(n\)-Super-HyperGraph as a time-labeled higher-order structure for dynamic hierarchical connectivity. The first contribution is a precise definition based on a finite base set \(V_0\), an \(n\)-level supervertex family \(V\subseteq \mathcal{P}^n(V_0)\), a superedge family \(E\subseteq \mathcal{P}^{\ast}(V)\), a time domain \(T\), and an activity map \(\Lambda:E\to 2^T\). The second contribution is a hierarchy result showing that static \(n\)-Super-HyperGraphs, temporal hypergraphs, and temporal graphs are recovered by forgetting time or by imposing natural restrictions on \(n\) and edge cardinality. The third contribution is a collection of structural results proving that snapshots, time restrictions, temporal unions, temporal intersections, activity complements, and time shifts preserve the defining conditions of the model. The paper also presents a construction algorithm from static snapshots, proves its correctness, analyzes its complexity, and illustrates the interpretation of the model through project-management, logistics, and smart-building examples. These results give a rigorous mathematical basis for studying dynamic higher-order systems in which both temporal activation and hierarchical identity are essential.

Let \(G=(V(G),E(G))\) be a finite, simple, undirected graph. For a vertex \(v\in V(G)\), the closed neighborhood is denoted by \(N_G[v]\) and consists of \(v\) together with every vertex adjacent to \(v\). A dominator coloring of \(G\) is a proper vertex coloring in which every vertex dominates at least one color class; equivalently, for each \(v\in V(G)\) there exists a color class \(C\) such that \(C\subseteq N_G[v]\). The least number of colors required in such a coloring is the dominator chromatic number, denoted by \(\chi_d(G)\). This manuscript determines the dominator chromatic number for the modular products \(P_n\diamond P_m\) and \(C_n\diamond C_m\), where \(P_n\) is a path and \(C_n\) is a cycle. The results give closed expressions in terms of \(h=\min\{n,m\}\) and \(g=\max\{n,m\}\), including the exceptional small orders where the parity pattern of the product changes. The constructions identify the color classes that are forced by proper coloring and the additional singleton classes needed to satisfy the domination condition. Representative colorings of \(P_5\diamond P_5\) and \(C_4\diamond C_6\) illustrate how the decisive vertices in the second row control the transition from ordinary proper coloring to dominator coloring.

Closed forms are derived for nested finite sums of the form \[\sum_{a_{n-1}=c}^{a_n}\sum_{a_{n-2}=c}^{a_{n-1}}\cdots\sum_{a_0=c}^{a_1}x^{a_0},\] where \(a_n\) and \(c\) are integers and \(x\) is real or complex. This elementary identity is then used to evaluate multiple sums whose summands contain terms of the Horadam sequence \(\bigl(W_j(a,b;p,q)\bigr)\). The sequence is defined by \[W_0=a,\qquad W_1=b,\qquad W_j=pW_{j-1}-qW_{j-2}\quad(j\geq 2),\] where \(a,b,p,q\in\mathbb C\) with \(p\ne0\) and \(q\ne0\). The resulting identities include weighted sums involving Lucas sequences of the first and second kinds, Fibonacci and Lucas numbers, gibonacci sequences, and products of two and three shifted terms. The formulas show how the depth of summation is absorbed into binomial coefficients and shifted sequence indices, yielding compact expressions suitable for direct use in recurrence and summation problems.

The first degcity index \(\operatorname{DC}_{1}(G)\) of a connected graph \(G\) is the edge sum \[\operatorname{DC}_{1}(G)=\sum\limits_{uv\in E(G)}\bigl[e_G(u)+e_G(v)\bigr]\bigl[d_G(u)+d_G(v)\bigr],\] where \(d_G(u)\) and \(e_G(u)\) denote the degree and eccentricity of a vertex \(u\), respectively. The index combines local valency and global distance information in a single degree–eccentricity descriptor. This paper determines closed expressions for the first degcity index under six standard graph operations: disjoint union, join, Cartesian product, composition, symmetric difference and disjunction. The formulas separate the contributions of edges inherited from the factor graphs from the contributions created by the operation. The statements use the eccentricity behaviour in joins and the edge and degree relations in product-type operations, giving formulas that are consistent with the usual definitions of these graph operations.

Let R be a ring with identity. The nil-clean graph of R is a graph, denoted by G_NC(R), whose vertex-set is the set R, and where two distinct vertices x and y are adjacent if and only if x + y is a nil-clean element of R. An element r ∈ R is called a nil-clean element if it can be decomposed a sum of an idempotent and a nilpotent element of R. Let G be a finite undirected graph. An automorphism φ of G is a permutation on the vertex-set V(G) such that the graph preserves adjacency, that is, φ(v₁) is adjacent to φ(v₂) if and only if v₁ is adjacent to v₂. The set of all automorphisms of G together with the composition operation of permutations forms the automorphism group of G. In this paper, we firstly compute the order of the automorphism groups of nil-clean graphs for the ring ℤ_n. And then we determine the structure of the automorphism groups of G_NC(ℤ_n) for n = p^k, pq, 2^kp^l, where p, q are distinct primes and k, l are positive integers.