The binding number of a graph quantifies how strongly its vertices are tied together by their neighborhoods. In this paper, we prove that if the binding number $b > 1$, then every isolated realizing set is an isolated dominating set. We further establish several inequalities relating the binding number and the isolated domination number, and derive bounds involving structural parameters and spectral quantities. Our results highlight a close interplay between domination-type parameters and the binding number.
In a graph $G = (V, E)$, a subset $S \subseteq V$ is called a dominating set if every vertex in $V \setminus S$ is adjacent to at least one vertex in $S$. For each vertex $u \in S$, the number of edges between $u$ and vertices in $V \setminus S$ is defined as the out-degree of $u$ with respect to $S$. A dominating set $S$ is termed an out-degree equitable restrained dominating set if $S$ is a restrained dominating set, and every vertex outside $S$ is adjacent to a vertex in $S$ with the degree difference at most one. In this paper, we investigate fundamental properties, derive bounds, and present key results regarding out-degree equitable restrained domination in graphs.
A method for finding the global minimum of continuous functions is proposed. The method is based on representing the optimized parameters as approximations by trigonometric polynomials, which makes it possible to approximate functions of arbitrary complexity. The method imposes no restrictions on the objective function whatsoever - in particular, it does not require knowledge of the Lipschitz constant. The method has been tested on numerous model problems with dimensionality up to 5000 variables.
Graph theory is a fundamental field of study across various scientific disciplines, including mathematics, chemistry, security, natural sciences, and computer science. It is extensively used to solve numerous real-world problems, particularly within network theory and security frameworks. Let G = (V(G, E(G)) be a molecular graph. Then in chemical graph theory, various graph parameters, known as topological indices, have been employed to establish correlations between molecular architectures and their chemical behavior, physical properties, or biological effects. These indices are also utilized to evaluate the robustness of graph-based structures. The double hexagonal chain graphs, denoted by D-n, represent a class of polycyclic aromatic hydrocarbons. In this paper, six distinct topological indices based on reverse degrees such as the reverse Nirmala index RN(D-n), the reverse geometric-arithmetic index RGA(D-n), the reverse inverse sum index ISI(D-n), the reverse atombond connectivity index RABC(D-n), the reverse forgotten index RF(D-n) and the reverse harmonic index RH(D-n) have been computed for the double hexagonal chain graphs. Furthermore, the results have been presented via both graphical representations and detailed numerical data. Then, the relation RF(D-n) > RN(D-n) > RGA(D-n)> RH(D-n) > ISI(D-n) > RABC(D-n) is obtained as the value of n increases.
This paper presents a structural analysis of multilayer network chain interactions, extending existing results beyond cycle polynomial formulations. The network is constructed by sequentially connecting disjoint graphs through complete bipartite links between consecutive layers. We characterize the distance structure, showing that interlayer distances depend only on the relative positions of layers, while intralayer distances are bounded by the interaction mechanism. Based on this, explicit expressions for the diameter, eccentricity, and radius are derived. In addition, bounds for vertex connectivity are established, highlighting the role of intermediate layers in network robustness and control. These results provide further insight into the structural behavior of multilayer networks and complement existing studies on graph polynomials.
The coarse deg-centric graph of a graph G, denoted by G(cd), is the graph with V(G(cd)) = V(G)and E(G(cd)) = {ViVj : d(G) (V-i , V-j) > deg (G)(v(i)). This transformation of the graph is called coarse deg-centrication of the graph. In this paper, we study the domination and domination number of coarse deg-centric graphs. Furthermore, we explore their fundamental properties and structural characteristics, offering insights into their behavior under various graph operations.
A prime labeling of a graph G with p vertices is a bijection f : V(G) -> {1, 2,..., p} such that gcd(f(x), f(y)) = 1 for every edge xy is an element of E(G). Prime labeling has been widely studied for several well-known graph families. In this paper, we establish prime labeling for graphs obtained from the disjoint union of two copies of helm graph under various edge subdivision operations. These include subdivisions of edges incident to the apex vertex, edges of the cycle C-n, edges joining the cycle to pendant vertices, and the barycentric subdivision of the helm graph.
A frequency assignment model is an algebraically configured unitary addition Cayley graph $\text {Cay}_ \text{UA} (R, \Omega)$ associated with a commutative ring $R$, denoted by $\Gamma_\text{UA}(R)$, where $\Omega$ is a generating symmetrical subset of $R$. A sharp upper frequency bound $\lambda_{2,1}$ of $\Gamma_\text{UA}(R)$ is easily accessible by imposing the $L(1,2)$-labeling function $g:V(\Gamma^{c}_{\text {UA}}(R)) \to [0, \infty)$ such that $|g(u)-g(v)| \ge 1$ if $d(u,v)=1$ and $|g(u)-g(v)| \to 2$ if $d(u,v)=2$ for a distance $d$ between any two vertices $u,v$ of $\Gamma^{c}_{\text {UA}}(R)$, where $\Gamma^{c}_{\text {UA}}(R)$ is the complement structure of $\Gamma_\text{UA}(R)$. In this paper, we have obtained $\lambda_{2,1}$ for $\text{Cay}_\text{UA} (R, \Omega)$ within $2 \Delta$. Also, it declares that the $L(2,1)$-labeling bound for any graph $G$ is less than $\Delta^2$ as well as exact minimum bound which was proposed by Griggs and Yeh.
In this paper, we developed a posynomial model for estimating the cost of green buildings in the South-East Nigeria. Green building is in line with the building philosophy of the inhabitants of the old South-East Nigeria. We collected secondary data on the cost of materials for a four-bedroom bungalow green building to test the workability of the model. The model accounts for the multi-objective nature of the component parts of the building. We obtained the Pareto optimal cost of constructing such a building to be N8,148,335.00 and determined the dimensions (length, width, and height) of the building.
In this paper, we study the total co-independent domination (TC-ID) number of various classes of graphs and their Cartesian products. We determine the TC-ID number for Cartesian products of complete graphs taken finitely many times, as well as for the Cartesian product of a complete graph with an arbitrary connected graph. Furthermore, we obtain additional results on the TC-ID number.
A graph G is said to be T-regular if for any clique Q of G and any vertex v outside Q, there exists a cover C of Q in G, and an edge e incident with v such that e is not incident with any vertices of V(C). In this paper, we characterize T-regular graphs and study their behaviour under graph operations and graph products. Also, we derive sufficient conditions for join of two graphs, Cartesian, tensor and strong products of two graphs to be T-regular.
A topological index is a numerical invariant of a graph, typically defined in terms of the degrees or distances of its vertices, and is widely used in QSAR and QSPR studies. In this paper, we introduce the diminished downhill Sombor index together with its exponential variant. Exact values of the proposed index are determined for regular graphs and several standard graph classes, including cycle graphs, complete graphs, path graphs, complete bipartite graphs, and star graphs. Corresponding closed-form expressions for the exponential are also derived. Furthermore, sharp upper and lower bounds for the diminished downhill Sombor index are established.
Let S(V, E) be a semigraph with no (m, e) -vertices [3]. If v(1), v(2), v(3),..., v(n) are the end vertices of S(V, E), then the ev-adjacency matrix of S(V, E), denoted by A(ev), is the n x n matrix A(ev) = [a(ij)], where a(ij) = {(2mij)(1) if v(i) and v(J) are adjacent 0 otherwise, where m(ij) is the number of middle vertices on the edge with end vertices v(i) and v(j). The ev-energy of a semigraph is the sum of absolute values of eigenvalues of the ev-adjacency matrix. The present paper also attempts to introduce some other definitions such as ev-walk, ev-graph, ev-complete semigraphs and ev-connected semigraphs. Apart from that, some results related to these definitions are discussed. As a semigraph is a generalization of a graph, the results pertaining to the adjacency matrix of a graph can also be generalized.
A total coloring of a graph G = (V, E) entails the meticulous assignment of colors to both vertices and edges, ensuring that neither two adjacent vertices nor two adjacent edges receive the same color. Furthermore, it requires that for each edge, the colors of its end-points and the edge itself must differ. The total chromatic number of a graph G, indicated as x"(G), signifies the minimum number of colors required for such a total coloring. The renowned conjecture by Behzad and Vizing asserts that for any graph G, the inequality 0(G) + 1 <= x"(G) <= 0(G) + 2 holds, where 0(G) signifies the graph's maximum degree. In this paper, we obtain that odd line graphs of odd complete graphs f(Kn), where n = 2 yv + 1 and yv >= 3 (with yv being odd), can be adorned with a total coloring with 0(f(Kn )) + 1 colors.
In this paper, we study some palindromic properties in the prefixes of the Hall-Thue word (an infinite ternary square-free word) $$u_{H T}=a b c a c b a b c b a c a b c a c b a c a b c b a b c a c b a b c \ldots$$ which is a fixed point of the Hall morphism $\varphi_H: a \mapsto a b c$, $b \mapsto a c, \quad c \mapsto b$. First, we get a factorization of $\varphi_H^n(a)$, for all $n \geq 1$, via its longest bi-occurrent prefix which we describe a priori. Then, we study the structure of the palindromes in $\varphi_H^n(a)$ and deduce its palindromic defect as a function of $n$. Lastly, we get an explicit formula for the palindromic defect of the prefixes of the infinite word $u_{H T}$.
The placement of renewable energy nodes in a smart grid can be studied as a finite covering problem on a graph whose vertices represent substations, load buses, candidate renewable units and control-capable switching points. This paper develops a domination-theoretic model for renewable-node placement under admissibility, target-load and outage-tolerance constraints. Given a graph $G=(V,E)$, an admissible placement set $Q\subseteq V$ and a target set $T\subseteq V$, a renewable dominating set is a subset $D\subseteq Q$ whose closed neighbourhood covers $T$. The model is extended to $k$-renewable domination and $(k,t)$-resilient renewable domination, where every target vertex is covered by at least $k+t$ selected vertices before failures and therefore retains k-fold coverage after any t selected-node outages. Feasibility criteria, monotonicity relations, failure-tolerant coverage, backbone-connectivity and NP-hardness results are proved. A greedy marginal-gain algorithm is also presented for constructive screening, and a small benchmark computation on the stated feeder abstraction is included to compare exact and greedy placements. Graph illustrations show how ordinary domination, redundant domination and connected resilient placement differ in a smart-grid feeder. The proposed framework supplies a discrete mathematical layer for renewable energy integration, microgrid formation and resilience-oriented smart-grid planning.
An incidence of a graph G is a pair (x, e), where x is a vertex of G and e is an edge of G incident to x. Two incidences (x, e) and (y, f) are adjacent if any one of the following holds: (i) x = y, or (ii) e = f, or (iii) xy = e or f. In [3], Brualdi and Massey introduced the concept of incidence coloring of a graph G as a mapping from the set of incidences of G to a finite set of colors such that adjacent incidences receive distinct colors. A signed graph (G, sigma) consists of a graph G and the signature sigma : E(G) -> {+1, -1}. In this paper, we define an incidence coloring of signed graphs as a natural generalization of the usual notion of incidence coloring of unsigned graphs. We prove that our definition is compatible with switching operation. We also prove that the incidence chromatic number (in signed sense) of a signed graph (G, sigma)coincide with the incidence chromatic number (in the usual unsigned sense) of its underlying graph G. The exact value or upper bounds which are known for the incidence chromatic numbers of some well-known families of unsigned graphs are also mentioned for their signed versions, namely, signed cycles, signed trees, signed complete graphs and signed toroidal grids.
A set $S \subseteq V$ of vertices in a graph $G=(V, E)$ is called a dominating set if every vertex in $V \backslash S$ is adjacent to a vertex in $S$. An independent transversal dominating set is defined as a dominating set which intersects every maximum independent set in $G$. The minimum cardinality of an independent transversal dominating set is called the independent transversal domination number of $G$ and is denoted by $\gamma_{i t}(G)$. The domination contraction number of a graph $G$ is defined as the minimum number of edges that must be contracted in order to decrease the domination number and it is denoted by $c t_\gamma(G)$. We extend this idea of domination contraction number to independent transversal domination. The independent transversal domination contraction number of a graph $G$ denoted by $c t_{\gamma_{i t}}(G)$ is the minimum number of edges that must be contracted in order to decrease the independent transversal domination number. In this paper, we initiate the study of this parameter.
The k-step domination problem is to find a minimum vertex set D c V of a graph G = (V, E) such that every vertex of the graph is either in D or at exact distance k from some vertex of D. In this paper, we extend this concept and initiate the study of L-step domination problem, which is to find a minimum vertex set D c V of a graph G = (V, E) such that each vertex v of the graph G is either in D or at exact distance av from some vertex of D, where av is an arbitrary nonnegative integer assigned to v. We show that the L-step domination problem is NP-complete for many known classes of graphs. Then we compute the L-step domination numbers for some special classes of graphs and for a special list L. Finally, by using a labeling method, we provide a linear time algorithm to produce an L-step dominating set of a tree.
Topological indices are numerical values associated with the molecular structure of chemical compounds that help in understanding their physicochemical, biological, and structural properties. We present a new graph metric called the general elliptic Sombor index, denoted as Ea, b(G)for a graph G. This index is calculated by summing (eta u+ eta v ) ( eta 2 across a + eta 2 )b u v all edges eta eta vuin G, where a and b are real numbers, and u eta represents the degree of the vertex u. We compute this index for several standard classes of graphs and establish upper and lower bounds involving the general sum connectivity index, the general Sombor index, and extremal vertex degrees. Additionally, we develop Nordhaus-Gaddum type results for this index and demonstrate its application in modeling physicochemical properties of benzenoid hydrocarbons.
