Petersen Graph 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.

Structure of some ( [formula omitted], [formula omitted])-free graphs with application to colorings

We use Pt and Ct to denote a path and a cycle on t vertices, respectively. A diamond is a graph obtained from two triangles that share exactly one edge. A kite is a graph that consists of a diamond and another vertex adjacent to a vertex of degree 2 of the diamond. A gem is a graph that consists of a P4 plus a vertex adjacent to all vertices of the P4. Choudum et al. (2021) proved that every (P7,C7,C4, gem)-free graph has either a clique cutset or a vertex whose neighbors is the union of two cliques unless it is a clique blowup of the Petersen graph, and every non-Petersen (P7,C7,C4, diamond)-free graph has either a clique cutset or has a vertex of degree at most max{2,ω(G)−1}. In this paper, we extend these two results respectively to (P7,C4, gem)-free graphs and (P7,C4, diamond)-free graphs. We also prove that if G is a (P7,C4, kite)-free graph with δ(G)≤ω(G) and without clique cutsets, then there is an integer ℓ≥0 such that G=Kℓ+H, where H is either the Petersen graph or a well-defined graph F. This implies a polynomial algorithm which can determine the chromatic number of (P7,C4, diamond)-free graphs, color (P7,C4, kite)-free graphs G with (ω(G)+1) colors, and color (P7,C4, gem)-free graphs G with (2ω(G)−1) colors.

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).

Turán problems in pseudorandom graphs

AbstractGiven a graph $F$ , we consider the problem of determining the densest possible pseudorandom graph that contains no copy of $F$ . We provide an embedding procedure that improves a general result of Conlon, Fox, and Zhao which gives an upper bound on the density. In particular, our result implies that optimally pseudorandom graphs with density greater than $n^{-1/3}$ must contain a copy of the Peterson graph, while the previous best result gives the bound $n^{-1/4}$ . Moreover, we conjecture that the exponent $1/3$ in our bound is tight. We also construct the densest known pseudorandom $K_{2,3}$ -free graphs that are also triangle-free. Finally, we give a different proof for the densest known construction of clique-free pseudorandom graphs due to Bishnoi, Ihringer, and Pepe that they have no large clique.

On cyclically 4-connected cubic graphs

A 3-connected cubic graph is cyclically 4-connected if it has at least n ≥ 8 vertices and when removal of a set of three edges results in a disconnected graph, only one component has cycles. By introducing the notion of cycle spread to quantify the distance between pairs of edges, we get a new characterization of cyclically 4-connected graphs. Let Q n and V n denote the ladder and Möbius ladder on n ≥ 8 vertices, respectively. We prove that a 3-connected cubic graph G is cyclically 4-connected if and only if G is either the Petersen graph, Q n or V n for n ≥ 8 , or G is obtained from Q 8 or Q 10 by bridging pairs of edges with cycle spread at least ( 1 , 2 ) . The concept of cycle spread also naturally leads to methods for constructing cyclically k-connected cubic graphs from smaller ones, but for k ≥ 5 the method is not exhaustive.

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.

Fault tolerability evaluation for the component edge-connectivity of folded Petersen networks

Analysis on the link fault tolerability of multiprocessor systems plays a significance role in designing and optimizing for multiprocessor systems. The component edge-connectivity, as a generalization of the classical edge-connectivity, is a considerable measurement for the analysis of the robustness of multiprocessor systems. The k-component edge-connectivity of a connected graph G, expressed by cλk(G), is the minimum cardinality of an edge set whose removal is disconnected and has at least k components in the resulting graph. The n-th cartesian product of the Petersen graph P, called as the n-dimensional folded Petersen network Pn, is one of the most appealing topologies for multiprocessor systems. In this paper, cλk(Pn) for 1≤k≤2n−1 are obtained.

Removable and forced subgraphs of graphs

A connected graph G is H-removable if H is a subgraph of G, G has at least |V(H)|+2 vertices and G−V(H′) has a perfect matching for every subgraph H′ of G isomorphic to H. So a connected graph is kK2-removable if and only if it is k-extendable, where kK2 is a matching of size k. Further G is an H-forced graph if G−V(H′) has a unique perfect matching for every subgraph H′ of G isomorphic to H. In this paper, we first characterize kK2-forced graphs for k≥2 as K2k+2 and Kk+1,k+1 by using randomly matchable graphs. Let Pi denote a path with i vertices. We show that P3- and P4-removable graphs are factor-critical and 1-extendable respectively. By using ear decomposition, we obtain that a connected graph G is P3-forced if and only if G is K5 or C2k+1 (k≥2). Finally we prove that a connected graph G is P4-forced if and only if G is isomorphic to K6, K3,3, C2k (k≥3), the Petersen graph, a uniform theta graph Θ3n (n≥3) or C12+.

Some conditions for hamiltonian cycles in 1-tough (K2 ∪ kK1)-free graphs

Let k≥2 be an integer. We say that a graph G is (K2∪kK1)-free if it does not contain K2∪kK1 as an induced subgraph. Recently, Shi and Shan conjectured that every 1-tough and 2k-connected (K2∪kK1)-free graph is hamiltonian. In this paper, we solve this conjecture by proving that every 1-tough and k-connected (K2∪kK1)-free graph with minimum degree at least 3(k−1)2 is hamiltonian or the Petersen graph.

Optimal radio labelings of the Cartesian product of the generalized Peterson graph and tree

A radio labeling of a graph [Formula: see text] is a function [Formula: see text] such that [Formula: see text] for every pair of distinct vertices [Formula: see text] of [Formula: see text]. The radio number of [Formula: see text], denoted by [Formula: see text], is the smallest number [Formula: see text] such that [Formula: see text] has radio labeling [Formula: see text] with max[Formula: see text] = [Formula: see text]. In this paper, we give a lower bound for the radio number for the Cartesian product of the generalized Petersen graph and tree. We present two necessary and sufficient conditions, and three other sufficient conditions to achieve the lower bound. Using these results, we determine the radio number for the Cartesian product of the Peterson graph and stars.

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.