Family Of Graphs Research Articles

Modified Entire Forgotten Topological Index of Graphs: A Theoretical and Applied Perspective

Topological indices are numerical invariants derived from graph structures that are essential tools used in computational chemistry and biology for encoding molecular information. By exploiting the inherent symmetries of molecular graphs, we develop efficient algorithms to compute these indices, particularly for large and complex molecules. These indices are rooted in vertex degrees, edge degrees, and other graph parameters, have been extensively studied, and are crucial for understanding the relationship between molecular structure and properties. Recent research has focused on the entire Zagreb indices, which integrate both vertex and edge degrees considering adjacency and incidence relationships. This paper introduces a novel variant, namely, the modified forgotten entire Zagreb index. The efficacy of this new index is underscored by its robust correlation with the physical and chemical properties of octane isomers and lower benzenoid hydrocarbons. Additionally, we derive explicit formulas for this index for several significant graph families.

Exploring modular multiplicative divisor labeling to expand graph families

The graph [Formula: see text] is a trivial graph consisting of a single vertex with no edges. In contrast, the graph [Formula: see text] is a complete bipartite graph with one internal node and [Formula: see text] leaf vertices. The join of two graphs [Formula: see text] and [Formula: see text] (with central vertex [Formula: see text]), represented as [Formula: see text], is a graph including vertices [Formula: see text] and edges [Formula: see text]. When two graphs, [Formula: see text] and [Formula: see text] are joined, the result is a graph in which vertex in graph [Formula: see text] is linked to every vertex in graph [Formula: see text]. Modular multiplicative divisor (MMD) labeling is a vertex and edge labeling scheme with the following key features: Vertex labeling: MMD labeling establishes a bijection between the vertices of the graph [Formula: see text] and the natural numbers from 1 to [Formula: see text]. This bijection ensures a one-to-one correspondence, providing a unique label for each vertex. Edge labeling: The labeling of edges follows a specific rule, the edge’s label is determined by calculating the result of multiplying the labels assigned to its connected vertices, with the outcome adjusted by modulo [Formula: see text]. We demonstrate the modular multiplication labeling when combining two graphs through their join (assuming [Formula: see text] is even) and also on the even arbitrary super subdivision (EASS) of the join of two graphs. Additionally, we explore a related research question that arises in this particular context. The findings have potential applications in network topology design and optimization.

Directed capacity-preserving subgraphs: hardness and exact polynomial algorithms

We introduce and discuss the Minimum Capacity-Preserving Subgraph (MCPS) problem: given a directed graph with edge capacities cap and a retention ratio α∈(0,1), find the smallest subgraph that, for each pair of vertices (u, v), preserves at least a fraction α of a maximum u-v-flow’s value. This problem originates from the practical setting of reducing the power consumption in a computer network: it models turning off as many links as possible, while retaining the ability to transmit at least α times the traffic compared to the original network. First we prove that MCPS is NP-hard already on a restricted set of directed acyclic graphs (DAGs) with unit edge capacities. Our reduction also shows that a closely related problem (which only considers the arguably most complicated core of the problem in the objective function) is NP-hard to approximate within a sublogarithmic factor already on DAGs. In terms of positive results, we present two algorithms that solve MCPS optimally on directed series-parallel graphs (DSPs): a simple linear-time algorithm for the special case of unit edge capacities and a cubic-time dynamic programming algorithm for the general case of non-uniform edge capacities. Further, we introduce the family of laminar series-parallel graphs (LSPs), a generalization of DSPs that also includes cyclic and very dense graphs. Their properties allow us to solve MCPS on LSPs by employing our DSP-algorithms as subroutines. In addition, we give a separate quadratic-time algorithm for MCPS on LSPs with unit edge capacities that also yields straightforward quadratic time algorithms for several related problems such as Minimum Equivalent Digraph and Directed Hamiltonian Cycle on LSPs.

On the (S 2)-condition of edge rings for cactus graphs

A cactus graph is a connected graph in which every block is either an edge or a cycle. In this paper, we will examine cactus graphs where all the blocks are 3-cycles, i.e., triangular cactus graphs, of diameter 4. Our main focus is to prove that the corresponding edge ring of this family of graphs is not normal and satisfies Serre’s condition ( S 2 ) . We use a criterion due to Katthän for non-normal affine semigroup rings.

Unique fault-tolerance resolvability codes for certain generalized convex polytopes with applications

For a non-trivial connected graph [Formula: see text], an ordered subset [Formula: see text] of vertices [Formula: see text] any pair of different vertices [Formula: see text], if [Formula: see text] for some [Formula: see text]. Such a set [Formula: see text] is said to be a resolving set (RS) for [Formula: see text] and the smallest cardinality of [Formula: see text] is called the [Formula: see text] [Formula: see text] of [Formula: see text]. A RS [Formula: see text] for [Formula: see text] is said to be fault-tolerant resolving set (FTRS) if the property of resolving [Formula: see text] holds by [Formula: see text] for every [Formula: see text] in [Formula: see text]. The minimum cardinality of such a set [Formula: see text] is called the fault-tolerant metric dimension (FTMD) of [Formula: see text]. It is generally denoted by [Formula: see text]. In this paper, the notion of FTRS and that of FTMD have been determined for two complex families of infinite convex polytope graphs, which are obtained by joining certain copies of the prism graph together with an antiprism graph. For these two families of planar graph (viz., [Formula: see text] & [Formula: see text]), we investigate several properties as well as we compare their metric and FTMD.

Fuzzy Vertex Range Labeling of Some Graph Families

Fuzzy Vertex Range Labeling of Some Graph Families

Optimization in Symmetric Trees, Unicyclic Graphs, and Bicyclic Graphs with Help of Mappings Using Second Form of Generalized Power-Sum Connectivity Index

The topological index (TI), sometimes referred to as the connectivity index, is a molecular descriptor calculated based on the molecular graph of a chemical compound. Topological indices (TIs) are numeric parameters of a graph used to characterize its topology and are usually graph-invariant. The generalized power-sum connectivity index (GPSCI) for the graph is ΩYα(Ω)=∑ζϱ∈E(Ω)(dΩ(ζ)dΩ(ζ)+dΩ(ϱ)dΩ(ϱ))α, while the second form of the GPSCI is defined as Yβ(Ω)=∑ζϱ∈E(Ω)(dΩ(ζ)dΩ(ζ)×dΩ(ϱ)dΩ(ϱ))β. In this paper, we investigate Yβ in the family of trees, unicyclic graphs, and bicyclic graphs. We determine optimal graphs in the desired families for Yβ using certain mappings. For graphs with maximal values, two mappings are used, namely A and B, while for graphs with minimal values, mapping C and mapping D are considered.

On spectrum of Sombor matrix and Sombor energy of graphs

Abstract The Sombor index ( SO {\mathrm{SO}} ) is a recently introduced degree-based graph invariant, defined as the sum over all pairs of adjacent vertices u , v {u,v} of the term d u 2 + d v 2 {\sqrt{{d_{u}^{2}+d_{v}^{2}}}} , where d u {d_{u}} and d v {d_{v}} denote the degrees of vertices u and v, respectively. The matrix associated with SO {\mathrm{SO}} is the Sombor matrix, and its spectrum is the Sombor spectrum. In this paper, the connected graphs having exactly two and exactly three Sombor eigenvalues are characterized. Bounds are obtained for the spectral radius and energy of the Sombor matrix, and the corresponding extremal graphs are determined. In addition, the Sombor spectra of several families of graphs are calculated.

Graceful Local Antimagic Labeling of Graphs: A Pattern Analysis Using Python

Graph labeling is the process of assigning labels to vertices and edges under certain conditions. This paper investigates the graceful local antimagic labeling of various graph families, excluding symmetric labelings, using computational experiments and Python-based algorithms. Through these experiments, we identify new results and patterns within specific graph classes. The study expands on the existing literature by offering computational evidence, proposing algorithms for the verification of labelings, and exploring the relationship between the local antimagic labeling and the chromatic number. Our results increase the understanding of graph labeling and offer insights into its computational aspects.

Spanning Trees Entropy of some of the families of graphs Based on a triangle

The number of spanning trees is an important quantity characterizing the reliability of a network. In this paper, we find the explicit formulas of the number of spanning trees of some new of the families of sequence graphs generated by triangle graph with its special feature in iteration. Using the electrically equivalent transformations, we obtain the weights of corresponding equivalent graphs and we further derive relationships for spanning trees between these graphs and transformed graphs. Finally, we compare the entropy of our graphs together and with other studied graphs of average degree.

Saturation of \(K_{4}\) Subdivisions in Multidimensional Grids

Let \(\mathcal{F}\) be a family of graphs, and \(H\) a ``host'' graph. A spanning subgraph \(G\) of \(H\) is called \(\mathcal{F}\)- saturated in \(H\) if \(G\) contains no member of \(\mathcal{F}\) as a subgraph, but \(G+e\) contains a member of \(\mathcal{F}\) for any edge \(e\in E(H) - E(G)\). We let \(Sat(H,\mathcal{F})\) be the minimum number of edges in any graph \(G\) which is \(\mathcal{F}\)-saturated in \(H\), where \(Sat(H,\mathcal{F}) = |E(H)|\) if \(H\) contains no member of \(\mathcal{F}\) as a subgraph. Let \(P_{m}^{r}\) be the \(r\)-dimensional grid, with entries in each coordinate taken from \(\{1,2,\cdots , m\}\), and \(K_{t}\) the complete graph on \(t\) vertices. Also let \(S(F)\) be the family of all subdivisions of a graph \(F\). There has been substantial previous work on extremal questions involving subdivisions of graphs, involving both \(Sat(K_{n},S(F))\) and the Turan function \(ex(K_{n},S(F))\), for \(F = K_{t}\) or \(F\) a complete bipartite graph. In this paper we study \(Sat(H, S(F))\) for the host graph \(H = P_{m}^{r}\), and \(F = K_{4}\), motivated by previous work on \(Sat(K_{n}, S(K_{t}))\). Our main results are the following; 1) If at least one of \(m\) or \(n\) is odd with \(m\geq 5\) and \(n\geq 5\), then \(Sat(P_{m}\times P_{n}, S(K_{4})) = mn + 1.\) 2) For \(m\) even and \(m\geq 4\), we have \(m^{3} + 1 \le Sat(P_{m}^{3}, S(K_{4}))\le m^{3} + 2.\) 3) For \( r\geq 3\) with \(m\) even and \(m\geq 4\), we have \(Sat(P_{m}^{r}, S(K_{4})) \le m^{r} + 2^{r-1} - 2\).

D-irregularity Strength of a Graph

We initiate to study a \(D\)-irregular labeling, which generalizes both non-inclusive and inclusive \(d\)-distance irregular labeling of graphs. Let \(G=(V(G),E(G))\) be a graph, \(D\) a set of distances, and \(k\) a positive integer. A mapping \(\varphi\) from \(V(G)\) to the set of positive integers \(\{1,2,\dots,k\}\) is called a \(D\)-irregular \(k\)-labeling of \(G\) if every two distinct vertices have distinct weights, where the weight of a vertex \(x\) is defined as the sum of labels of vertices whose distance from \(x\) belongs to \(D\). The least integer \(k\) for which \(G\) admits a \(D\)-irregular labeling is the \(D\)-irregularity strength of \(G\) and denoted by \(\mathrm{s}_D(G)\). In this paper, we establish several fundamental properties on \(D\)-irregularity strength for arbitrary graphs. We also determine this parameter exactly for families of graphs with small diameter or small maximum degree.

Further Results on Equitable Near Proper Coloring of Derived Graph Families

A proper coloring assigns distinct colors to the adjacent vertices of a graph. An equitable near proper coloring of a graph G is an improper coloring in which neighbouring vertices are allowed to receive the same color such that the cardinalities of two distinct color classes differ by not more than one and the number of monochromatic edges is minimised by giving certain restrictions on the number of color classes that can have an edge between them. This paper discusses the equitable near proper coloring of line, middle, and total graphs of certain graph classes, such as paths, cycles, sunlet graphs, star graphs, and gear graphs.

Analyzing Pathway Energy And Time Complexity In Flower Families Through Vertex-Disjoint Paths

This research delves into the pathway energy framework for flower families, a class of simple connected graphs, whose path matrix p is constructed such that each entry pij quantifies the maximum number of vertex-disjoint paths. By analyzing the characteristic values of this matrix, we establish the pathway energy bounds specific to these flower graph families. Additionally, a comprehensive algorithm is developed to evaluate the time complexity across different flower family configurations, utilizing numerous trials to capture their average, maximum, and minimum computational behaviors. This analysis offers a comparative study of the structural intricacies that lead to increased computational complexity, highlighting which graph topologies tend to impose higher algorithmic challenges. The proposed method introduces a refined and adaptable approach, deepening the exploration of characteristic graph properties and their computational impact, thereby expanding the practical applications of these findings in graph theory.

On Bridge Graphs with Local Antimagic Chromatic Number 3

Let G=(V,E) be a connected graph. A bijection f:E→{1,…,|E|} is called a local antimagic labeling if, for any two adjacent vertices x and y, f+(x)≠f+(y), where f+(x)=∑e∈E(x)f(e), and E(x) is the set of edges incident to x. Thus, a local antimagic labeling induces a proper vertex coloring of G, where the vertex x is assigned the color f+(x). The local antimagic chromatic number χla(G) is the minimum number of colors taken over all colorings induced by local antimagic labelings of G. In this paper, we present some families of bridge graphs with χla(G)=3 and give several ways to construct bridge graphs with χla(G)=3.

On the skewness of the generalized Heawood graphs

By the skewness of a graph, we mean the minimum number of its edges whose deletion results in a planar graph. We determine the skewness of a large family of cubic bipartite graphs (which includes the Heawood graph as a special case). Moreover, we also determine those classes of these cubic graphs which are π -skew whose resulting plane graphs (upon deleting the right minimum number of edges) are hexagulations.

Synthetic graphs for link prediction benchmarking

Abstract Predicting missing links in complex networks requires algorithms that are able to explore statistical regularities in the existing data. Here we investigate the interplay between algorithm efficiency and network structures through the introduction of suitably-designed synthetic graphs. We propose a family of random graphs that incorporates both micro-scale motifs and meso-scale communities, two ubiquitous
structures in complex networks. A key contribution is the derivation of theoretical upper bounds for link prediction performance in our synthetic graphs, allowing us to estimate the predictability of the task and obtain an improved assessment of the performance of any method. Our results on the performance of classical methods (e.g., Stochastic Block Models, Node2Vec, GraphSage) show that the performance of all methods correlate with the theoretical predictability, that no single method is universally superior, and that each of the methods exploit different characteristics known to exist in large classes of networks. Our findings underline the need for careful consideration of graph structure when selecting a link prediction method and emphasize the value of comparing performance against synthetic benchmarks. We provide open-source code for generating these synthetic graphs, enabling further research on link
prediction methods.

On Generalised Majority Edge-Colourings of Graphs

A $\frac{1}{k}$-majority $l$-edge-colouring of a graph $G$ is a colouring of its edges with $l$ colours such that for every colour $i$ and each vertex $v$ of $G$, at most $\frac{1}{k}$'th of the edges incident with $v$ have colour $i$. We conjecture that for every integer $k\geq 2$, each graph with minimum degree $\delta\geq k^2$ is $\frac{1}{k}$-majority $(k+1)$-edge-colourable and observe that such result would be best possible. This was already known to hold for $k=2$. We support the conjecture by proving it with $2k^2$ instead of $k^2$, which confirms the right order of magnitude of the conjectured optimal lower bound for $\delta$. We at the same time improve the previously known bound of order $k^3\log k$, based on a straightforward probabilistic approach. As this technique seems not applicable towards any further improvement, we use a more direct non-random approach. We also strengthen our result, in particular substituting $2k^2$ by $(\frac{7}{4}+o(1))k^2$. Finally, we provide the proof of the conjecture itself for $k\leq 4$ and completely solve an analogous problem for the family of bipartite graphs.

Further study on the second status connectivity index of graphs

The second status connectivity index (SSC index, for short) of a simple connected graph A is defined as the sum of the terms σ A ( a ) σ A ( b ) over all edges ab of A , where σ A ( a ) , the status of a vertex a in A , is the sum of the distances between a and all other vertices of A . The aim of the paper is to study the SSC index through investigating its behavior under various families of composite graphs and a number of mathematico-chemically interesting graphs derived from them.

Grim Under a Compensation Variant

Games on graphs are a well studied subset of combinatorial games. When analyzing a game, heuristics and strategies for winning are often at the forefront of the discussion. One such combinatorial graph game that can be considered is Grim. In Grim there are winning strategies for a variety of well-known families of graphs, many of which favor the first player. Hoping to develop a fairer Grim, we look at Grim played under a slightly different rule set and develop winning strategies for this modified version of the game. Throughout, we compare our new results to those previously known and discuss whether our altered Grim is a fairer game than the original.