Generalized Petersen Graphs Research Articles

The crossing number of the generalized Petersen graph P(3k,k) in the projective plane

ABSTRACT The crossing number of a graph G in a surface Σ, denoted by c r Σ ( G ) , is the minimum number of pairwise intersections of edges in a drawing of G in Σ. Let k be an integer satisfying k ≥ 3 , the generalized Petersen graph P ( 3 k , k ) is the graph with vertex set V ( P ( 3 k , k ) ) = { u i , v i ∣ i = 1 , 2 , … , 3 k } and edge set E ( P ( 3 k , k ) ) = { u i u i + 1 , u i v i , v i v k + i ∣ i = 1 , 2 , … , 3 k } , the subscripts are read modulo 3 k . In this paper we investigate the crossing number of P ( 3 k , k ) in the projective plane. We determine the exact value of c r N 1 ( P ( 3 k , k ) ) is k–2 when 3 ≤ k ≤ 7 , moreover, for k ≥ 8 , we get that k − 2 ≤ c r N 1 ( P ( 3 k , k ) ) ≤ k − 1.

On Distance-Balanced Generalized Petersen Graphs

On Distance-Balanced Generalized Petersen Graphs

On Spanning Tree Packing Number of the Complement of Generalized Petersen Graph and Cocktail Party Graph

For any graph G, the spanning tree packing number of \(\sigma\) (G), is the maximum number of edge-disjoint spanning trees contained in G. In this study, we determined the maximum number of edge-disjoint spanning trees of the generalized petersen graph and cocktail graph.

Binary programming formulations for the upper domination problem

We consider Upper Domination, the problem of finding the minimal dominating set of maximum cardinality. Very few exact algorithms have been described for solving Upper Domination. In particular, no binary programming formulations for Upper Domination have been described in literature, although such formulations have proved quite successful for other kinds of domination problems. We introduce two such binary programming formulations, and show that both can be improved with the addition of extra constraints which reduce the number of feasible solutions. We compare the performance of the formulations on various kinds of graphs, and demonstrate that (a) the additional constraints improve the performance of both formulations, and (b) the first formulation outperforms the second in most cases, although the second performs better for very sparse graphs. Also included is a short proof that the upper domination number of any generalized Petersen graph P(n, k) is equal to n.

Locally irregular edge-coloring of subcubic graphs

A graph is locally irregular if no two adjacent vertices have the same degree. A locally irregular edge-coloring of a graph G is such an (improper) edge-coloring that the edges of any fixed color induce a locally irregular subgraph. Among the graphs admitting a locally irregular edge-coloring, i.e., decomposable graphs, only one is known to require 4 colors, while for all the others it is believed that 3 colors suffice. In this paper, we prove that decomposable claw-free graphs with maximum degree 3, all cycle permutation graphs, and all generalized Petersen graphs admit a locally irregular edge-coloring with at most 3 colors. We also discuss when 2 colors suffice for a locally irregular edge-coloring of cubic graphs and present an infinite family of cubic graphs of girth 4 which require 3 colors.

On 2-Rainbow Domination of Generalized Petersen Graphs P(ck,k)

We obtain new results on the 2-rainbow domination number of generalized Petersen graphs P(ck,k). Exact values are established for all infinite families where the general lower bound 45ck is attained. In all other cases lower and upper bounds with small gaps are given.

Fuzzy Quotient -3 Cordial Labeling on Generalized Petersen Graph

Objectives: To analyze the existence of fuzzy quotient-3 cordial labeling on the generalized Petersen graph. Methods: The method involves mathematically defining how to label the vertex of a generalized Petersen graph and demonstrating that these formulations produce fuzzy quotient-3 cordial labeling. Findings: In this study, we proved that the generalized Petersen graph GP(h;m) ; 1  m < ⌊ h 2 ⌋ is fuzzy quotient-3 cordial, except for h  0( mod 6). Novelty: Here, we give fuzzy quotient-3 cordial labeling to some families of graphs, namely Durer graph, Desargues graph, Dodecahedral graph, Mobius Kantor graph, Petersen graph, Cubical graph, Cubic symmetric graph, Nauru graph and the generalized Petersen graph GP(h;m) . Keywords: Labeling; Fuzzy Quotient -3 Cordial; Petersen Graph; Generalized Petersen Graph; Mobius Kantor Graph

Wiener polarity and Wiener index of double generalized Petersen graph

A topological index is a numerical parameter of a graph which characterizes some of the topological properties of the graph. The concepts of Wiener polarity index and Wiener index were established in chemical graph theory by means of the distances. The double generalized Petersen graph denoted by DP(n,k) is obtained by attaching the vertices of outer pendent vertices to inner pendent vertices lying at distance k. The length of the outer and inner cycle is n, thus the number of vertices are 4n and the number of edges in the DP(n,k) are 6n. In this paper, the Wiener polarity index of DP(n,k for 3≤n≤6 and for n≥6k+1 is computed. Further, the Wiener index of DP(n,k), for k={1,2} is determined.

Cyclic base ordering of generalized Petersen graphs

A cyclic base ordering (CBO) of a connected graph [Formula: see text] is a cyclic ordering of [Formula: see text] such that every cyclically consecutive [Formula: see text] edge induces a spanning tree of [Formula: see text]. The density of [Formula: see text] is defined to be [Formula: see text]; and [Formula: see text] is uniformly dense if [Formula: see text] for every connected subgraph [Formula: see text] of [Formula: see text]. It was conjectured by Kajitani, Ueno and Miyano that [Formula: see text] has a CBO if and only if [Formula: see text] is uniformly dense. In this paper, we study CBO of generalized Petersen graphs to support this conjecture, and we prove that [Formula: see text] and [Formula: see text] have CBO.

Under military war weapon support the economic bond level estimation using generalized Petersen graph with imputation

Several countries of the world are involved in mutual and collaborative business of military equipments, weapons in terms of their production, sales, technical maintenance, training and services. As a consequence, manufacturing of boms, rockets, missiles and other ammunitions have taken structured and smooth shape to help others where and when needed. Often the military support among countries remain open for information to the media, but sometime remain secret due to the national security and international political pressure. Such phenomenon (hidden or open support ) is a part of military supply chain and could be modeled like a Petersen graph considering vertices as countries and edges as economic bonds. For a large graphical structure, without sampling, it is difficult to find out average economic bonding (open &amp; secret) between any pair of countries involved in the military business or support. This paper presents a sample based estimation methodology for estimating the mean economic bond value among countries involved in the military support or business. Motivation to the problem is derived from current Russia-Ukraine war situation and a kind of hidden support to war by NATO countries. A node sampling procedure is proposed whose bias, mean-squared error and other properties are derived. Results are supported with empirical studies. Findings are compared with particular cases and confidence intervals are used as a basic tool of comparison. Pattern imputation is used together with a new proposal of CIImputation method who has been proved useful for filling the missing value, specially when secret economic support data from involved countries found missing. The current undergoing war between Ukraine and Russia and secret weapon, economic support from NATO countries is an application of the proposed methodology contained in this paper. Key words: Graph, Petersen Graph, Estimator, Bias, Mean Squared Error (MSE), Optimum Choice, Confidence intervals (CI), Nodes (vertices), Pattern Imputation, CI-Imputation (LLimputation and UL-imputation), Economic Bonds, Military War, Weapon Support

On the 2-rainbow independent domination numbers of some graphs

By suitably adjusting the tropical algebra technique we compute the rainbow independent domination numbers of several infinite families of graphs including Cartesian products C_n Box P_m and C_n Box C_m for all n and mle 5, and generalized Petersen graphs P(n, 2) for n ge 3.

On alpha -Excellent Graphs

A graph G is alpha -excellent if every vertex of G is contained in some maximum independent set of G. In this paper, we characterize alpha -excellent bipartite graphs, alpha -excellent unicyclic graphs, alpha -excellent simplicial graphs, alpha -excellent chordal graphs, alpha -excellent block graphs, and we show that every generalized Petersen graph is alpha -excellent.

Cyclic coverings of graphs. Counting rooted spanning forests and trees, Kirchhoff index, and Jacobians

The purpose of this survey is to describe invariants of cyclic coverings of graphs. The covered graph is assumed to be fixed, and the cyclic covering group has an arbitrarily large order. A classical example of such a covering is a circulant graph. It covers a one-vertex graph with a prescribed number of loops. More sophisticated objects representing the family of cyclic coverings include $I$-, $Y$-, and $H$-graphs, generalized Petersen graphs, sandwich-graphs, discrete tori, and many others. We present analytic formulae for counting rooted spanning forests and trees in cyclic coverings, establish their asymptotics, and study the arithmetic properties of these quantities. Moreover, in the case of circulant graphs we give exact formulae for computing the Kirchhoff index and present structural theorems for the Jacobians of such graphs. Bibliography: 95 titles.

Spectrum of prism graph and relation with network related quantities

&lt;abstract&gt;&lt;p&gt;Spectra of network related graphs have numerous applications in computer sciences, electrical networks and complex networks to explore structural characterization like stability and strength of these different real-world networks. In present article, our consideration is to compute spectrum based results of generalized prism graph which is well-known planar and polyhedral graph family belongs to the generalized Petersen graphs. Then obtained results are applied to compute some network related quantities like global mean-first passage time, average path length, number of spanning trees, graph energies and spectral radius.&lt;/p&gt;&lt;/abstract&gt;

Injective edge chromatic index of generalized Petersen graph P(ck,k)

An injective [Formula: see text]-edge coloring of a graph [Formula: see text] is a [Formula: see text]-edge coloring [Formula: see text] of [Formula: see text] such that [Formula: see text] for any three consecutive edges [Formula: see text] and [Formula: see text] of a path or a 3-cycle. The injective edge chromatic index of [Formula: see text], denoted by [Formula: see text], is the minimum [Formula: see text] such that [Formula: see text] has an injective [Formula: see text]-edge coloring. Li and Chen determined the injective edge chromatic index of the generalized Petersen graph [Formula: see text] for [Formula: see text]. In this paper, we consider the injective edge coloring of the generalized Petersen graph [Formula: see text] and show that [Formula: see text] for [Formula: see text].

Maximizing the number of independent sets in claw-free cubic graphs

Let G be a graph. An independent set of G is a subset of vertices no two of which are connected by an edge. Denote by I(G) the set of all independent sets of G. The independence polynomial of G is I(G;λ)=∑I∈I(G)λ|I|.The cartesian product C3□K2 is commonly called the triangular-prism, it is also a generalized Petersen graph on 6 vertices and denoted by GP(3,1). In this paper, we use the occupancy method to show that for any claw-free cubic graph G,|I(G)|≤|I(GP(3,1))||V(G)|/6,furthermore, for any λ∈(0,1541],I(G;λ)≤(I(GP(3,1);λ)|V(G)|/6,with equality holding in both cases if and only if G is a disjoint union of copies of GP(3,1).

Counting rooted spanning forests for circulant foliation over a graph

In this paper, we present a new method to produce explicit formulas for the number of rooted spanning forests $f(n)$ for the infinite family of graphs $H_n=H_n(G_1,G_2,\ldots,G_m)$ obtained as a circulant foliation over a graph $H$ on $m$ vertices with fibers $G_1,G_2,\ldots,G_m.$ Each fiber $G_i=C_n(s_{i,1},s_{i,2},\ldots,s_{i,k_i})$ of this foliation is the circulant graph on $n$ vertices with jumps $s_{i,1},s_{i,2},\ldots,s_{i,k_i}.$ This family includes the family of generalized Petersen graphs, $I$-graphs, sandwiches of circulant graphs, discrete torus graphs and others. The formulas are expressed through Chebyshev polynomials. We prove that the number of rooted spanning forests can be represented in the form $f(n)=p f(H)a(n)^2,$ where $a(n)$ is an integer sequence and $p$ is a prescribed natural number depending on the number of odd elements in the set of $s_{i,j}.$ Finally, we find an asymptotic formula for $f(n)$ through the Mahler measure of the associated Laurent polynomial.

The Alternative of Sensor Placement in Multi-Story Buildings through the Metric Dimension Approach: A Representation of Generalized Petersen Graphs

In a public facility or private office where many people can get together, a fire detection device is a mandatory tool as an emergency alarm in the facility. However, the expense of the installation of the device is a troublesome matter. So, optimization is needed to minimize the number of these devices. The way to implement is to select the appropriate position to place the devices in public facilities. The research discussed the placement of the sensors in multi-story buildings. The multi-story buildings could be represented as cube composition graphs with the number of rooms, and the connectivity between the floor and its rooms was equal. The concept of this multi-story building was modeled into a generalized Petersen graph where a vertex represented a room, and an edge was the connectivity of rooms. The basis obtained on that metric dimension was represented as a sensor placed on the building. Then, the optimization of device placement was seen as determining the metric dimensions of the Petersen graph. In the research, the alternative sensor placements were computed using the graph metric dimension approach implemented in Python. The research successfully implements the metric dimension of to using Python code to obtain the alternative of its basis. A basic alternative indicates the location of the device placement like fire detectors, network access points, or other sensors inside a building.

On Three-Rainbow Dominationof Generalized Petersen Graphs P(ck,k)

New results on singleton rainbow domination numbers of generalized Petersen graphs P(ck,k) are given. Exact values are established for some infinite families, and lower and upper bounds with small gaps are given in all other cases.

Positive matching decompositions of graphs

A matching M in a graph Γ is positive if Γ has a vertex-labeling such that M coincides with the set of edges with positive weights. A positive matching decomposition (pmd) of Γ is an edge-partition M 1 , … , M p of Γ such that M i is a positive matching in Γ − ( M 1 ∪ ⋯ ∪ M i − 1 ) , for i = 1 , … , p . The pmds of graphs are used to study algebraic properties of the Lovász–Saks–Schrijver ideals arising from orthogonal representations of graphs. We give a characterization of pmds of graphs in terms of alternating closed walks and apply it to study pmds of various classes of graphs including complete multipartite graphs, (regular) bipartite graphs, cacti, generalized Petersen graphs, etc. We further show that computation of pmds of a graph can be reduced to that of its maximum pendant-free subgraph.