Subdivision Graph Research Articles

Approximation algorithms for maximum weighted internal spanning trees in regular graphs and subdivisions of graphs

Abstract Let $G$ be a vertex-weighted connected graph of $n$ vertices and let $T$ be a spanning tree of $G$. We call $T$ a maximum weighted internal spanning tree of $G$ if the sum of the weights of the internal vertices of $T$ is the maximum over all spanning trees of $G$. The maximum weighted internal spanning tree (MaxwIST) problem asks to find such a spanning tree $T$ of $G$. The problem is NP-hard. We give an $O(dn)$ time approximation algorithm for $d$-regular graphs of $n=|V|$ vertices that computes a spanning tree with total weight of the internal vertices is at least $\frac{\beta _{d}}{\beta _{d} +d-2} - \epsilon $ of the total weight of all the vertices of the graph for any $\epsilon>0$, where $\beta _{d} = (d-1)H_{d-1}$, and $H_{d-1} = \sum _{i=1}^{d-1} i^{-1}$ is the $(d-1)$th harmonic number. For every $d \geq 3$ and $n_{0} \geq 1$, we show the construction of a $d$-regular graph of at least $n_{0}$ vertices, such that for any of its spanning trees, $\frac{w(I)}{w(V)}\le \frac{d}{d+1}$ holds. We give an $O(dn)$ time approximation algorithm for subdivisions of $d$-regular graphs, where the ratio of the internal weight of the spanning tree with the total vertex weight of the graph is at least $\frac{d-1}{2d-3} - \epsilon $ for $\epsilon>0$. We extend our study to $x$-subdivisions of Hamiltonian and hypoHamiltonian graphs, where each edge of the original Hamiltonian or hypoHamiltonian graph has been subdivided at least $x$ times. For those two graph classes, we show that there exists a spanning tree with internal vertex weight at least $1-\frac{2}{x-1}$ of the total vertex weight of the graph. Furthermore, we give $O(n)$ time algorithm for $x$-subdivisions of biconnected outerplanar graphs and $4$-connected planar graphs to achieve the above bound.

Interval colourable orientations of graphs

A graph is interval colourable if it admits a proper edge colouring in which for each vertex the set of colours of edges that are incident with the vertex is an interval of integers. An oriented graph is interval colourable if it admits a proper arc colouring in which for each vertex the set of colours of in-arcs and the set of colours of out-arcs that are incident with the vertex are both intervals of integers. In this paper we apply a classical approach and a new special trail technique to confirm the existence of interval colourable orientations of graphs from three classes: i) all k-trees with Δ(G)≤2k and k∈{2,3,4} and all k-paths with arbitrary k∈N; ii) graphs in which the length of every even closed trail is at least 2s and each such a trail has common edges with no more than d other even closed trails and 21−2se(d+1)≤1; iii) graphs that are decomposable into a bipartite interval colourable graph and a graph whose each connected component has at most one cycle that, if it exists, has odd length. Consequently, cactus graphs, graphs that are homeomorphic to Halin graphs, full subdivision graphs and others have interval colourable orientations.

Distance Spectra of Some Double Join Operations of Graphs

In literature, several types of join operations of two graphs based on subdivision graph, Q-graph, R-graph, and total graph have been introduced, and their spectral properties have been studied. In this paper, we introduce a new double join operation based on H1,H2-merged subdivision graph. We compute the spectrum of a special block matrix and then use it to describe the distance spectra of some double join operations of graphs. At last, we give several families of distance equienergetic graphs of diameter 3.

The Second Omega Index

Background: The omega index has been recently introduced to identify a variety of topological and combinatorial aspects of a graph with a new viewpoint. As a continuing study of the omega index, by considering the incidence of edges and vertices to the adjacency of the vertices, in this paper, we have introduced the second omega index Ω2 and then computed it over some well-known graph classes. Methods: Many combinatorial counting methods have been utilized in the proofs. The edge partition is frequently used throughout the work. Naturally, some graph theoretical lemmas are also used. Results: In particular, trees having the smallest and largest Ω2 have been constructed. Finally, the second omega index of some derived graphs, such as line graphs, subdivision graphs, and vertex-semitotal graphs, has been presented. Conclusion: Omega invariant has already been explored in many papers. It has been defined in terms of vertex degrees. Vertices correspond to the atoms in a molecule and a calculation, which only depends on the atomic parameters, is not even comparable with a calculation containing both atoms and chemical bonds between them. With this idea in mind, we have evaluated some mathematical properties of the second omega index, which has great potential in chemical applications related to the number of cycles in the molecular graph.

Isospectral graphs via spectral bracketing

In this article, we develop a perturbative technique to construct families of non-isomorphic discrete graphs which are isospectral for the standard (also called normalised) Laplacian and its signless version. We use vertex contractions as a graph perturbation and spectral bracketing with auxiliary graphs which have certain eigenvalues with high multiplicity. There is no need to know explicitly the eigenfunctions of the corresponding graphs. In principle, one only needs to know the multiplicity of the eigenvalues of the auxiliary graphs, and that these eigenvalues are all different. We illustrate the method by presenting several families of examples of isospectral graphs including fuzzy balls, complete bipartite graphs and subdivision graphs obtained from the previous examples. All the examples constructed turn out to be also isospectral for the standard (Kirchhoff) Laplacian on the associated equilateral metric graph.

A Study on Degree Based Topological Indices of Harary Subdivision Graphs With Application

Combinatorial design theory and graph decompositions play a critical role in the exploration of combinatorial design theory and are essential in mathematical sciences. The process of graph decomposition involves partitioning the set of edges in a graph G. An n-sun graph, characterized by a cycle with an edge connecting each vertex to a terminating vertex of degree one, is introduced in this study. The concept of n-sun decomposition is applied to certain even-order graphs. The indices covered in this study include the general connectivity index of the harary graphs, Zagreb indices, symmetric division degree indices and randic indices.

Computation of wiener polynomial and index of line subdivision friendship and line subdivision bifriendship graphs using matlab program

A topological index is a branch of chemical graph theory that is vital to analyzing the physio-chemical characteristics of chemical compound structures divided into a degree-based molecular structure such as Zagreb indices, a distance-based molecular structure such as Wiener index, and a mixed such as Gutman index. In this paper, some definitions, results, and examples of Wiener polynomial and index for subdivision graph of friendship, bifriendship graphs, line subdivision graph of friendship, and bifriendship graphs were introduced. Moreover, we used the MATLAB program to calculate the Wiener polynomial and index of these graphs and refer to some applications.

Eigenvalues of the generalized subdivision graph with applications to graph energy

For a graph [Formula: see text] with vertex set [Formula: see text] and edge set [Formula: see text], let [Formula: see text] denotes the subdivision graph of [Formula: see text] with vertex set [Formula: see text]. In [Formula: see text], replace each vertex [Formula: see text], [Formula: see text], by [Formula: see text] vertices and join every vertex to the neighbors of [Formula: see text]. Then in the resulting graph, replace each vertex [Formula: see text], [Formula: see text], by [Formula: see text] vertices and join every vertex to the neighbors of [Formula: see text]. The resulting graph is denoted by [Formula: see text]. This generalizes the construction of the subdivision graph [Formula: see text] to [Formula: see text] of a graph [Formula: see text]. In this paper, we provide the complete information about the spectrum of [Formula: see text] using the spectrum of [Formula: see text]. Further, we determine the Laplacian spectrum of [Formula: see text] using the Laplacian spectrum of [Formula: see text], when [Formula: see text] is a regular graph. Also, we find the Laplacian spectrum of [Formula: see text] using the Laplacian spectrum of [Formula: see text] when [Formula: see text]. The energy of a graph [Formula: see text] is defined as the sum of the absolute values of the eigenvalues of [Formula: see text]. The incidence energy of a graph [Formula: see text] is defined as the sum of the square roots of the signless Laplacian eigenvalues of [Formula: see text]. Finally, as an application, we show that the energy of the graph [Formula: see text] is completely determined by the incidence energy of the graph [Formula: see text].

A revisit to the planar subdivision graph: free space detection in a dynamic environment with polygonal obstacles

A revisit to the planar subdivision graph: free space detection in a dynamic environment with polygonal obstacles

Broadcast Domination in Line Graphs of Trees

A dominating broadcast of a graph G is a function f : V ( G ) → { 0 , 1 , 2 , … , diam ( G ) } such that f ( v ) ⩽ e ( v ) for all v ∈ V ( G ) , where e ( v ) is the eccentricity of v , and for every vertex u ∈ V ( G ) , there exists a vertex v with f ( v ) > 0 and d ( u , v ) ⩽ f ( v ) . The cost of f is ∑ v ∈ V ( G ) f ( v ) . The minimum of costs over all the dominating broadcasts of G is called the broadcast domination number γ b ( G ) of G . A graph $G$ is said to be radial if γ b ( G ) = rad ( G ) . In this article, we give tight upper and lower bounds for the broadcast domination number of the line graph L ( G ) of G , in terms of γ b ( G ) , and improve the upper bound of the same for the line graphs of trees. We present a necessary and sufficient condition for radial line graphs of central trees, and exhibit constructions of infinitely many central trees T for which L ( T ) is radial. We give a characterization for radial line graphs of trees, and show that the line graphs of the i -subdivision graph of K 1 , n and a subclass of caterpillars are radial. Also, we show that γ b ( L ( C ) ) = γ ( L ( C ) ) for any caterpillar C .

Construction of Simultaneous Cospectral Graphs for Adjacency, Laplacian and Normalized Laplacian Matrices

In this paper we construct several classes of non-regular graphs which are co-spectral with respect to all the three matrices, namely, adjacency, Laplacian and normalized Laplacian, and hence we answer a question asked by Butler [?]. We make these constructions starting with two pairs (G1, H1) and (G2, H2) of A-cospectral regular graphs, then considering the subdivision graphs S(Gi) and R-graphs ℛ(Hi), i = 1, 2, and finally making some kind of partial joins between S(G1) and ℛ(G2) and S(H1) and ℛ(H2). Moreover, we determine the number of spanning trees and the Kirchhoff index of the newly constructed graphs.

The study of line graphs of subdivision graphs of some rooted product graphs via K-Banhatti indices

The degree-based topological indices are numerical graph invariants that are used to link a molecule’s structural characteristics to its physical, and chemical characteristics. In the investigation, and study of the structural features of a chemical network, it has emerged as one of the most potent mathematical techniques. In this paper, we study the degree-based topological invariants, called K-Banhatti indices, of the line graphs of some rooted product graphs namely, [Formula: see text], [Formula: see text], and ith vertex rooted product graph [Formula: see text] which are derived by the concept of subdivision.

Generalized entropies of subdivision-corona networks

The entropy of a graph is an information-theoretic quantity which expresses the complexity of a graph. Entropy functions have been used successfully to capture different aspects of graph complexity. The generalized graph entropies result from applying information measures to a graph using various schemes for defining probability distributions over the elements of the graph. In this paper, we investigate the complexity of a class of composite graphs based on subdivision graphs and corona product evaluating the generalized graph entropies, and we present explicit formulae for the complexity of subdivision-corona type product graphs.

Chemical Applicability of Second Order Sombor Index

In this paper, we introduce the higher-order Sombor index of a molecular graph. In addtition, we compute the second order Sombor index of some standard class of graphs and line graph of subdivision graph of 2D-lattice, nanotube and nanotorus of TUC_4C_8[p,q] and also we obtain the expressions of the second order Sombor index of the line graph of subdivision graph of tadpole graph, wheel graph, ladder graph and chain silicate network CSn. Further, we study the linear regression analysis of the second order Sombor index with the entropy, acentric factor, enthalpy of vaporization and standard enthalpy of vaporization of an octane isomers.

On Degree-Based Topological Indices of Petersen Subdivision Graph

In this paper, we adequately describe the generalised petersen graph, expanding to the categories of graphs. We created a petersen graph, which is cyclic and has vertices that are arranged in the centre and nine gons plus one vertex, leading to the factorization of regular graphs. Petersen graph is still shown in graph theory literature, nevertheless.

Resolvability in Subdivision Graph of Circulant Graphs

Circulant networks are a very important and widely studied class of graphs due to their interesting and diverse applications in networking, facility location problems, and their symmetric properties. The structure of the graph ensures that it is symmetric about any line that cuts the graph into two equal parts. Due to this symmetric behavior, the resolvability of these graph becomes interning. Subdividing an edge means inserting a new vertex on the edge that divides it into two edges. The subdivision graph G is a graph formed by a series of edge subdivisions. In a graph, a resolving set is a set that uniquely identifies each vertex of the graph by its distance from the other vertices. A metric basis is a resolving set of minimum cardinality, and the number of elements in the metric basis is referred to as the metric dimension. This paper determines the minimum resolving set for the graphs Hl[1,k] constructed from the circulant graph Cl[1,k] by subdividing its edges. We also proved that, for k=2,3, this graph class has a constant metric dimension.

On mixed metric dimension in subdivision, middle, and total graphs

Let G be a graph and let S(G), M(G), and T(G) be the subdivision, the middle, and the total graph of G, respectively. Let dim(G), edim(G), and mdim(G) be the metric dimension, the edge metric dimension, and the mixed metric dimension of G, respectively. In this paper, for the subdivision graph it is proved that max{dim(G), edim(G)} ≤ mdim(S(G)) ≤ mdim(G). A family of graphs Gn is constructed for which mdim(Gn ) − mdim(S(Gn )) ≥ 2 holds and this shows that the inequality mdim(S(G)) ≤ mdim(G) can be strict, while for a cactus graph G, mdim(S(G)) = mdim(G). For the middle graph it is proved that dim(M(G)) ≤ mdim(G) holds, and if G is tree with n 1(G) leaves, then dim(M(G)) = mdim(G) = n 1(G). Moreover, for the total graph it is proved that mdim(T(G)) = 2n 1(G) and dim(G) ≤ dim(T (G)) ≤ n 1(G) hold when G is a tree.

Odd Fibonacci edge irregular labeling for some trees obtained from subdivision and vertex identification operations

Let G be a graph with p vertices and q edges and be an injective function, where k is a positive integer. If the induced edge labeling defined by for each is a bijection, then the labeling f is called an odd Fibonacci edge irregular labeling of G. A graph which admits an odd Fibonacci edge irregular labeling is called an odd Fibonacci edge irregular graph. The odd Fibonacci edge irregularity strength ofes(G) is the minimum k for which G admits an odd Fibonacci edge irregular labeling. In this paper, the odd Fibonacci edge irregularity strength for some subdivision graphs and graphs obtained from vertex identification is determined.

Cops and robber on variants of retracts and subdivisions of oriented graphs (Brief Announcement)

Cops and Robber is one of the most studied two-player pursuit-evasion games played on graphs, where multiple cops, controlled by one player, pursue a single robber. The cop number of a graph is the minimum number of cops that can ensure the capture of the robber. In directed graphs, two kinds of moves are defined for players: strong move, where a player can move both along and against the orientation of an arc to an adjacent vertex; and weak move, where a player can only move along the orientation of an arc to an out-neighbor. We study three variants of Cops and Robber on oriented graphs: strong cop model, where the cops can make strong moves while the robber can only make weak moves; normal cop model, where both cops and the robber can only make weak moves; and weak cop model, where the cops can make weak moves while the robber can make strong moves. We study the cop number of these models with respect to several variants of retracts on oriented graphs and establish that the strong and normal cop number of an oriented graph remains invariant in their strong and distributed retracts, respectively. Next, we go on to study all three variants with respect to the subdivisions of graphs and oriented graphs. Finally, we establish that all these variants remain computationally difficult even when restricted to the class of 2-degenerate bipartite graphs.

Calculation of Gourava topological indices in HAC5C6C7[p, q] nanotubes

One of the best and most widely used carbon structures is carbon nanotubes. The most important features of carbon nanotubes are high durability, flexibility and good conductivity. In this article, the graph of HAC5C6C7[p, q] carbon nanotube structure is investigated using a section of its surface. By presenting a method for obtaining the degree of vertices in the relevant area, the topological indices of Gourava and Hyper Gourava are calculated, and the relevant theorems are expressed. The Gourava indices and Hyper Gourava indices are then calculated for the line graph and subdivision graph of HAC5C6C7[p, q] Also, to compare the obtained values, the diagrams of Gourava and Hyper Gourava indices are drawn.