Generalized Petersen Graphs Research Articles

SPANNING TREES IN $\mathbb {Z}$ -COVERS OF A FINITE GRAPH AND MAHLER MEASURES

Abstract Using the special value at $u=1$ of Artin–Ihara L-functions, we associate to every $\mathbb {Z}$ -cover of a finite connected graph a polynomial, which we call the Ihara polynomial. We show that the number of spanning trees for the finite intermediate graphs of such a cover can be expressed in terms of the Pierce–Lehmer sequence associated to a factor of the Ihara polynomial. This allows us to express the asymptotic growth of the number of spanning trees in terms of the Mahler measure of this polynomial. Specialising to the situation where the base graph is a bouquet or the dumbbell graph gives us back previous results in the literature for circulant and I-graphs (including the generalised Petersen graphs). We also express the p-adic valuation of the number of spanning trees of the finite intermediate graphs in terms of the p-adic Mahler measure of the Ihara polynomial. When applied to a particular $\mathbb {Z}$ -cover, our result gives us back Lengyel’s calculation of the p-adic valuations of Fibonacci numbers.

Explicit construction of mixed dominating sets in generalized Petersen graphs

Explicit construction of mixed dominating sets in generalized Petersen graphs

The conjecture on distance-balancedness of generalized Petersen graphs holds when internal edges have jumps 3 or 4

The conjecture on distance-balancedness of generalized Petersen graphs holds when internal edges have jumps 3 or 4

Local Metric Resolvability of Generalized Petersen Graphs

Local Metric Resolvability of Generalized Petersen Graphs

On an infinite family of integral Cayley graphs of Pauli groups

The classical Pauli group can be obtained as the central product of the dihedral group of 8 elements with the cyclic group of order 4. Inspired by this characterization, we introduce the notion of central product of Cayley graphs, which allows to regard the Cayley graph of a central product of groups as a quotient of the Cartesian product of the Cayley graphs of the factor groups. We focus our attention on the Cayley graph Cay(Pn,SPn) of the generalized Pauli group Pn on n-qubits; in fact, Pn may be decomposed as the central product of finite 2-groups, and a suitable choice of the generating set SPn allows us to recognize the structure of central product of graphs in Cay(Pn,SPn).Using this approach, we are able to recursively construct the adjacency matrix of Cay(Pn,SPn) for each n≥1, and to explicitly describe its spectrum and the associated eigenvectors. It turns out that Cay(Pn,SPn) is a (3n+2)-regular bipartite graph on 4n+1 vertices, and it has integral spectrum. This is a highly nontrivial property if one considers that, by choosing as a generating set for P1 the three classical Pauli matrices, one gets the so-called Möbius-Kantor graph, belonging to the class of generalized Petersen graphs, whose spectrum is not integral.

The Tenacity of Generalized Petersen Graphs

The Tenacity of Generalized Petersen Graphs

A new graph labeling with Tribonacci, Fibonacci and Triangular numbers

In this paper, a new graph labeling technique using three sequences of numbers, namely Tribonacci, Fibonacci and Triangular is introduced. They are named as Tribo-Fibo-Triangular labeling and denoted as TFΔ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta $$\\end{document} labeling. Assignment of the labeling is done based on the families of generalized Petersen graphs GP (n,2), GP(n,3) and a generalization of these graph are also presented. Applications related to recruitment of employees in a health care sector and the selection of teams in a school ambience are arrived by using of these three sequences of graphs along with computational intelligence. A board game is developed and presented using these three sequences.

The Locating Chromatic Number for the New Operation on Generalized Petersen Graphs N_P(m,1)

The Locating Chromatic Number for the New Operation on Generalized Petersen Graphs N_P(m,1)

The maximum genus of the generalized Petersen Graph, GP (<em>n, k</em>) for the cases <em>k</em> = 1, 2

In Topological graph theory, the maximum genus of graphs has been a fascinating subject. For a simple connected graph G, the maximum genus γM(G) is the largest genus of an orientable surface on which G has a 2-cell embedding. γM(G) has the upper bound, γM(G)≤[β/2], where β(G) denotes the Betti number and G is said to be upper embeddable if the equality holds. In this study, the maximum genus of GP(n, k) is established as γM(GP(n,k))=[(n+1)/2] for k = 1 and k = 2 by proving the upper embeddability of generalized Petersen graph, GP(n, k) for the cases k = 1 and k = 2. The proof is done by obtaining spanning trees T and examining the components in the edge complements GP(n, k)\T for the cases k = 1 and k = 2 of GP(n, k).

Generalized 3-Rainbow Domination in Graphs and Honeycomb System (HC(n))

In this paper, we introduce the concept of the generalized 3 -rainbow dominating function of a graph G . This function assigns an arbitrary subset of three colors to each vertex of the graph with the condition that every vertex (including its neighbors) must have access to all three colors within its closed neighborhood. The minimum sum of assigned colors over all vertices of G is defined as the g 3 -rainbow domination number, denoted by γ g 3 r . We present a linear-time algorithm to determine a minimum generalized 3-rainbow dominating set for several graph classes: trees, paths ( P n ) , cycles ( C n ) , stars ( K 1 , n ) , generalized Petersen graphs ( G P ( n , 2 ) , GP ( n , 3 ) ) , and honeycomb networks ( H C ( n ) ) .

Solving Edges Deletion Problem of Generalized Petersen Graphs

The dependability and effectiveness of a network can be investigated using a variety of graph-theoretic techniques, and the network's connectivity determines how reliable it is. Diameter in a network is often used to measure the efficiency of a network in the event that a network experiences issues such as a decline in the communication signal or a breakdown in the communication between its components. This paper examines the increase in diameter of the generalized Petersen graph after removing a certain number of edges. Finding the exact values of f(n,t) that represent the maximum diameter of an altered generalized Petersen graph GP(n,k,t) obtained after removing edges from for k=1 and t≥2.

Rainbow domination regular graphs that are not vertex transitive

Examples of graphs that are rainbow domination regular and not vertex transitive are given. This answers two questions asked in Kuzman (2020). We also characterize all generalized Petersen graphs that are 3-rainbow domination regular.

A note on bounds for the broadcast domination number of graphs

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 u∈V(G), there exists a vertex v with f(v)⩾1 and d(u,v)⩽f(v). The cost of f is ∑v∈V(G)f(v). The minimum cost over all the dominating broadcasts of G is called the broadcast domination number γb(G) of G. In this paper, we give new upper and lower bounds for γb(G). We show that both the bounds are tight. Also, we improve the upper bound for a subclass of regular graphs. Further, we explore the broadcast domination numbers of generalized Petersen graphs and a subclass of circulant graphs.

The locating chromatic number of generalized Petersen graphs with small order

It was conjectured by Asmiati (2018) that the generalized Petersen graph Pn,k has a locating chromatic number 4 if and only if (noddandk=1) or (n=4andk=2). In this paper, we give a negative answer to the conjecture posed by Asmiati. As a consequence, we are able to exhibit many counterexamples to the recent conjecture proposed, by proving that if (5≤n≤12) and (2≤k≤⌊n−12⌋) and (n,k)≠(12,5), then χLP(n,k)=4.

Beyond symmetry in generalized Petersen graphs

A graph is a core or unretractive if all its endomorphisms are automorphisms. Well-known examples of cores include the Petersen graph and the graph of the dodecahedron—both generalized Petersen graphs. We characterize the generalized Petersen graphs that are cores. A simple characterization of endomorphism-transitive generalized Petersen graphs follows. This extends the characterization of vertex-transitive generalized Petersen graphs due to Frucht, Graver, and Watkins and solves a problem of Fan and Xie. Moreover, we study generalized Petersen graphs that are (underlying graphs of) Cayley graphs of monoids. We show that this is the case for the Petersen graph, answering a recent mathoverflow question, for the Desargues graphs, and for the Dodecahedron—answering a question of Knauer and Knauer. Moreover, we characterize the infinite family of generalized Petersen graphs that are Cayley graphs of a monoid with generating connection set of size two. This extends Nedela and Škoviera’s characterization of generalized Petersen graphs that are group Cayley graphs and complements results of Hao, Gao, and Luo.

A single assignment of numbers for square divisor cordial and cube divisor cordial labelings on the generalized Petersen graphs GP (n,3)

In this paper, a work on the generalized Petersen graph GP [Formula: see text] is done, proving Generalized Petersen graph GP [Formula: see text] to be both Square Divisor Cordial Graphs and Cube Divisor Cordial Graphs by a single assignment of numbers on the same graph. Few conditions are provided to label the graphs in single assignment to prove two different labelings. A rule is developed to label the graphs and the results obtained are tabulated by actual verification for [Formula: see text] to [Formula: see text] but the rule holds good for all [Formula: see text] 4. Further, a few illustrations are provided to justify the rule given to label the graphs in a single assignment.

Totally antimagic total labeling of ladders, prisms and generalised Petersen graphs

A total labeling is called edge-antimagic total (vertex-antimagic total) if all edge-weights (vertex-weights) are pairwise distinct. If a labeling is simultaneously edge-antimagic total and vertex-antimagic total, it is called a totally antimagic total labeling. A graph that admits totally antimagic total labeling is called a totally antimagic total graph. In this paper, we prove that ladders, prisms and generalised Pertersen graphs are totally antimagic total graphs. We also show that the chain graph of totally antimagic total graphs is a totally antimagic total graphs.

\((d,1)\)-Total Labelling of Generalized Petersen Graphs \(P(n,k)\)

In this paper, we investigate the ( d , 1 ) -total labelling of generalized Petersen graphs P ( n , k ) for d ≥ 3 . We find that the ( d , 1 ) -total number of P ( n , k ) with d ≥ 3 is d + 3 for even n and odd k or even n and k = n 2 , and d + 4 for all other cases.

On graceful coloring of generalized Petersen graphs

The graphs used in this paper are simple, connected and finite graph. A graceful [Formula: see text]-coloring of a graph [Formula: see text] is a proper vertex coloring [Formula: see text], where [Formula: see text] which induces a proper edge coloring [Formula: see text] that defined by [Formula: see text]. Proper vertex coloring [Formula: see text] of a graph [Formula: see text] is a graceful coloring if [Formula: see text] is a graceful m-coloring for [Formula: see text]. The minimum vertex coloring, that can apply such that a graph [Formula: see text] can be colored with graceful coloring, is called a graceful chromatic number and it is denoted by [Formula: see text]. In this paper, we will investigate the graceful chromatic number of generalized Petersen graph [Formula: see text] for [Formula: see text].

Edge Resolvability in Generalized Petersen Graphs

The generalized Petersen graphs are a type of cubic graph formed by connecting the vertices of a regular polygon to the corresponding vertices of a star polygon. This graph has many interesting graph properties. As a result, it has been widely researched. In this work, the edge metric dimensions of the generalized Petersen graphs GP(2l + 1, l) and GP(2l, l) are explored, and it is shown that the edge metric dimension of GP(2l + 1, l) is equal to its metric dimension. Furthermore, it is proved that the upper bound of the edge metric dimension is the same as the value of the metric dimension for the graph GP(2l, l).