Subdivision Graph Research Articles

Radio Mean Graceful Graphs

A Radio Mean labeling of a simple, finite, undirected and connected graph G is a one to one map f:V(G) → N such that for two distinct vertices u and v of . The radio mean number of f, rmn(f), is the highest number assigned to any vertex of G. The radio mean number of G,rmn(G), is the minimum value of rmn(f), taken over all radio mean labelings of G. If rmn(G) = |V(G)|, we call such graphs as radio mea graceful. In this paper, we find the radio mean number of subdivision graph of complete graphs, Mongolian tent graphs, subdivision of friendship graphs and Diamond graphs and prove that these graphs are radio mean graceful.

SPN Graphs

A simple graph G is an SPN graph if every copositive matrix having graph G is the sum of a positive semidefinite and nonnegative matrix. SPN graphs were introduced in [N. Shaked-Monderer. SPN graphs: When copositive = SPN. Linear Algebra Appl., 509:82{113, 2016.], where it was conjectured that the complete subdivision graph of K4 is an SPN graph. This conjecture is disproved, which in conjunction with results in the Shaked-Monderer paper show that a subdivision of K_4 is a SPN graph if and only if at most one edge is subdivided. It is conjectured that a graph is an SPN graph if and only if it does not have an F_5 minor, where F_5 is the fan on five vertices. To establish that the complete subdivision graph of K_4 is not an SPN graph, rank-1 completions are introduced and graphs that are rank-1 completable are characterized.

Bounds on Inverse Sum Indeg Index of Subdivision Graphs

The inverse sum indeg index ISI(G) of a simple graph G is defined as the sum of the terms (d_G(u)d_G(v))/(d_G(u)+d_G(v)) over all edges uv of G, where d_G(u) denotes the degree of a vertex u of G. In this paper, we present several upper and lower bounds on the inverse sum indeg index of subdivision graphs and t-subdivision graphs. In addition, we obtain the upper bounds for inverse sum indeg index of S-sum, S_t-sum, S-product, S_t-product of graphs.

增强超立方体剖分图和全图的基尔霍夫指标

增强超立方体剖分图和全图的基尔霍夫指标

A class of highly symmetric graphs, symmetric cylindrical constructions and their spectra

In this article, we introduce the algebra of block-symmetric cylinders and we show that symmetric cylindrical constructions on base-graphs admitting commutative decompositions behave as generalized tensor products. We compute the characteristic polynomial of such symmetric cylindrical constructions in terms of the spectra of the base-graph and the cylinders in a general setting. This gives rise to a simultaneous generalization of some well-known results on the spectra of a variety of graph amalgams, as various graph products, graph subdivisions and generalized Petersen graph constructions. While our main result introduces a connection between spectral graph theory and commutative decompositions of graphs, we focus on commutative cyclic decompositions of complete graphs and tree-cylinders along with a subtle group labeling of trees to introduce a class of highly symmetric graphs containing the Petersen and the Coxeter graphs. Also, using techniques based on recursive polynomials we compute the characteristic polynomials of these highly symmetric graphs as an application of our main result.

Signless Laplacian spectrum of a class of generalized corona and its application

Let [Formula: see text] be a graph with [Formula: see text] edges, [Formula: see text] the subdivision graph of [Formula: see text] with [Formula: see text] the set of inserted vertices of [Formula: see text]. The generalized subdivision-edge corona graph [Formula: see text] of [Formula: see text] and [Formula: see text] is the graph obtained from [Formula: see text] and [Formula: see text] by joining the [Formula: see text]th vertex of [Formula: see text] to every vertex of [Formula: see text]. In this paper, we determine the [Formula: see text]-polynomial of the graph [Formula: see text]. Also, we construct infinitely many pairs of [Formula: see text]-cospectral graphs and compute the incidence energy of subdivision-edge corona graphs.

Hyperbolicity on Graph Operators

A graph operator is a mapping F : Γ → Γ ′ , where Γ and Γ ′ are families of graphs. The different kinds of graph operators are an important topic in Discrete Mathematics and its applications. The symmetry of this operations allows us to prove inequalities relating the hyperbolicity constants of a graph G and its graph operators: line graph, Λ ( G ) ; subdivision graph, S ( G ) ; total graph, T ( G ) ; and the operators R ( G ) and Q ( G ) . In particular, we get relationships such as δ ( G ) ≤ δ ( R ( G ) ) ≤ δ ( G ) + 1 / 2 , δ ( Λ ( G ) ) ≤ δ ( Q ( G ) ) ≤ δ ( Λ ( G ) ) + 1 / 2 , δ ( S ( G ) ) ≤ 2 δ ( R ( G ) ) ≤ δ ( S ( G ) ) + 1 and δ ( R ( G ) ) − 1 / 2 ≤ δ ( Λ ( G ) ) ≤ 5 δ ( R ( G ) ) + 5 / 2 for every graph which is not a tree. Moreover, we also derive some inequalities for the Gromov product and the Gromov product restricted to vertices.

Narumi–Katayama index of the subdivision graphs

ABSTRACTSubdivision is an important aspect in graph theory which allows one to calculate properties of some complicated graphs in terms of some easier graphs. Recently, the notion of r-subdivision was similarly defined as a quite useful generalization by adding r new vertices to each edge. Also the double graphs are used to ease some calculations, especially with the chemical graphs. Topological graph indices have become popular due to their applications in chemistry or related areas due to their advantages over time and money consuming laboratory experiments. In this paper, we calculate one of the topological graph indices, namely the Narumi–Katayama index of the subdivision, r-subdivision, double, subdivision of double and double of subdivision graphs of any given graph. We give some symmetry relations and results for the well-known graph classes.

3-total edge product cordial labeling of some new classes of graphs

In this paper we study 3-total edge product cordial (3-TEPC) labeling of some new classes of graphs. We study ladder graph, wheel graph, m isomorphic copies of n–cycle graph and dutch windmill graph in the context of 3-TEPC labeling. We also studied 3-TEPC labeling of some particular examples of graph operations like corona graphs and subdivision graphs.

First reformulated Zagreb indices of some classes of graphs

A topological index of a graph is a parameter related to the graph; it does not depend on labeling or pictorial representation of the graph. Graph operations plays a vital role to analyze the structure and properties of a large graph which is derived from the smaller graphs. The Zagreb indices are the important topological indices found to have the applications in Quantitative Structure Property Relationship(QSPR) and Quantitative Structure Activity Relationship(QSAR) studies as well. There are various study of different versions of Zagreb indices. One of the most important Zagreb indices is the reformulated Zagreb index which is used in QSPR study.
 In this paper, we obtain the first reformulated Zagreb indices of some derived graphs such as double graph, extended double graph, thorn graph, subdivision vertex corona graph, subdivision graph and triangle parallel graph. In addition, we compute the first reformulated Zagreb indices of two important transformation graphs such as the generalized transformation graph and generalized Mycielskian graph.

The harmonic index of edge-semitotal graphs, total graphs and related sums

For a connected graph G, there are several related graphs such as line graph L(G), subdivision graph S(G), vertex-semitotal graph R(G), edge-semitotal graph Q(G) and total graph T(G) [I. Gutman, B. Furtula, Ž. K. Vukićević and G. Popivoda, On Zagreb indices and coindices, MATCH Commun. Math. Comput. Chem. 74 (2015), 5-16, W. Yan, B.-Y. Yang and Y. -N. Yeh, The behavior of Wiener indices and polynomials of graphs under five graph decorations, Appl. Math. Lett. 20 (2007), 290-295]. Let F be one of symbols S, R, Q or T. The F-sum G1+FG2 of two connected graphs G1 and G2 is a graph with vertex set (V (G1) U E(G1)) × V (G2) in which two vertices (u1, v1) and (u2, v2) of G1 +F G2 are adjacent if and only if [u1 = u2 _ V (G1) and u1v2 _ E(G2)] or [v1 = v2 and u1u2 2 E(F(G))] [M. Eliasi and B. Taeri, Four new sums of graphs and their Wiener indices, Discrete Appl. Math. 157 (2009), 794-803]. In this paper, we investigate the harmonic index of edge-semitotal graphs, total graphs and F-sum of graphs, where F = Q or T.

Resistance Distance and Kirchhoff Index of the Corona-Vertex and the Corona-Edge of Subdivision Graph

The resistance distance is widely used in random walk, electronic engineering, and complex networks. One of the main topics in the study of the resistance distance is the computation problem. The subdivision graph $S(G)$ of a graph $G$ is the graph obtained by inserting a new vertex into every edge of $G$ . Two classes of new corona graphs, the corona-vertex of the subdivision graph $G_{1}\diamondsuit G_{2}$ and the corona-edge of the subdivision graph $G_{1}\star G_{2}$ , were defined by Lu and Miao. The adjacency spectrum and the signless Laplacian spectrum of the two new graph operations were computed when $G_{1}$ is an arbitrary graph and $G_{2}$ is an $r_{2}$ -regular graph. In this paper, we give the closed-form formulas for the resistance distance and Kirchhoff index of $G_{1}\diamondsuit G_{2}$ and $G_{1}\star G_{2}$ in terms of the resistance distance and Kirchhoff index of the factor graphs.

On the F-index and F-coindex of the line graphs of the subdivision graphs

The aim of this work is to investigate the F-index and F-coindex of the line graphs of the cycle graphs, star graphs, tadpole graphs, wheel graphs and ladder graphs using the subdivision concepts. F-index of the line graph of subdivision graph of square grid graph, 2D-lattice, nanotube and nanotorus of $TUC_{4}C_{8}[p, q]$ are also investigated here.

Anagram-Free Colorings of Graph Subdivisions

An anagram is a word of the form $WP$ where $W$ is a non-empty word and $P$ is a permutation of $W$. A vertex coloring of a graph is anagram-free if no subpath of the graph is an anagram. Anagram-f...

On the Laplacian Spectra of Some Double Join Operations of Graphs

Many variants of join operations of graphs have been introduced and their spectral properties have been studied extensively by many researchers. This paper mainly focuses on the Laplacian spectra of some double join operations of graphs. We first introduce the conception of double join matrix and provide a complete information about its eigenvalues and the corresponding eigenvectors. Further, we define four variants of double join operations based on subdivision graph, $Q$-graph, $R$-graph and total graph. Applying the result obtained for the double join matrix, we give an explicit complete characterization of the Laplacian eigenvalues and the corresponding eigenvectors of four variants in terms of the Laplacian eigenvalues and the eigenvectors of the factor graphs. These results generalize some well-known results about some join operations of graphs.

Topological Indices of the Line Graph of Subdivision Graph of Complete Bipartite Graphs

Topological Indices of the Line Graph of Subdivision Graph of Complete Bipartite Graphs

Topological Indices of the Total Graph of Subdivision Graphs

Topological Indices of the Total Graph of Subdivision Graphs

On the Second Order First Zagreb Index

Inspired by the chemical applications of higher-order connectivity index (or Randic index), we consider here the higher-order first Zagreb index of a molecular graph. In this paper, we study the linear regression analysis of the second order first Zagreb index with the entropy and acentric factor of an octane isomers. The linear model, based on the second order first Zagreb index, is better than models corresponding to the first Zagreb index and F-index. Further, we compute the second order first Zagreb index of line graphs of subdivision graphs of 2D-lattice, nanotube and nanotorus of TUC4C8[p; q], tadpole graphs, wheel graphs and ladder graphs.

More Results on Polygonal Sum Labeling of Graphs

Objectives: To explore and identify some new classes of graphs which exhibit polygonal sum labeling. Methods: In this article we use a methodology which fundamentally involves formulation and subsequent mathematical validation. Findings: Here we establish that the graphs – Star ( K 1,n ), Coconut Tree, Bistar (S m,n ), the Graph S m n k Comb, (P n K 1 ) and Subdivision graph S(K 1,n ) admit pentagonal, hexagonal, heptagonal, octagonal, nonagonal and decagonal sum labeling. Applications: One can explore to generalize these results and extend to give n-gonal labeling to some classes graphs. Sum labeling has already been used in the problems involving relational database management and hence one can try out to use polygonal sum labeling as well in these problems.

English

English