Path Pn Research Articles

Maximum energy bicyclic graphs containing two odd cycles with one common vertex

The energy of a graph is the sum of the absolute values of all eigenvalues of its adjacency matrix. Let Pn6,6 be the graph obtained from two copies of C6 joined by a path Pn−10. In 2001, Gutman and Vidović (2001) conjectured that the bicyclic graph with the maximal energy is Pn6,6. This conjecture is true for bipartite bicyclic graphs. For non-bipartite bicyclic graphs, Ji and Li (2012) proved the conjecture for bicyclic graphs which have exactly two edge-disjoint cycles such that one of them is even and the other is odd. This paper is to prove the conjecture for bicyclic graphs containing two odd cycles with one common vertex.

The Secure Metric Dimension of the Globe Graph and the Flag Graph

Let G = (V, E) be a connected, basic, and finite graph. A subset T=u1,u2,…,uk of V(G) is said to be a resolving set if for any y ∈ V(G), the code of y with regards to T, represented by CTy, which is defined as CTy=du1,y,du2,y,…,duk,y, is different for various y. The dimension of G is the smallest cardinality of a resolving set and is denoted by dim(G). If, for any t ∈ V – S, there exists r ∈ S such that S–r∪t is a resolving set, then the resolving set S is secure. The secure metric dimension of 𝐺 is the cardinal number of the minimum secure resolving set. Determining the secure metric dimension of any given graph is an NP-complete problem. In addition, there are several uses for the metric dimension in a variety of fields, including image processing, pattern recognition, network discovery and verification, geographic routing protocols, and combinatorial optimization. In this paper, we determine the secure metric dimension of special graphs such as a globe graph Gln, flag graph Fln, H- graph of path Pn, a bistar graph Bn,n2, and tadpole graph T3,m. Finally, we derive the explicit formulas for the secure metric dimension of tadpole graph Tn,m, subdivision of tadpole graph ST3,m, and subdivision of tadpole graph STn,m.

Ramsey numbers of cliques versus monotone paths

One formulation of the Erdős-Szekeres monotone subsequence theorem states that for any red/blue coloring of the edge set of the complete graph on {1,2,…,N}, there exists a monochromatic red s-clique or a monochromatic blue increasing path Pn with n vertices, provided N>(s−1)(n−1). Here, we prove a similar statement as above in the off-diagonal case for triple systems, with the quasipolynomial bound N>2c(logn)s−1. For the tth power Pnt of the ordered increasing graph path with n vertices, we prove a near linear bound cn(logn)s−2 which improves the previous bound that applied to a more general class of graphs than Pnt due to Conlon-Fox-Lee-Sudakov.

On local distance antimagic labeling of graphs

Let G = ( V , E ) be a graph of order n and let f : V → { 1 , 2 … , n } be a bijection. For every vertex v ∈ V , we define the weight of the vertex v as w ( v ) = ∑ x ∈ N ( v ) f ( x ) where N(v) is the open neighborhood of the vertex v. The bijection f is said to be a local distance antimagic labeling of G if w ( u ) ≠ w ( v ) for every pair of adjacent vertices u , v ∈ V . The local distance antimagic labeling f defines a proper vertex coloring of the graph G, where the vertex v is assigned the color w(v). We define the local distance antimagic chromatic number χ l d ( G ) to be the minimum number of colors taken over all colorings induced by local distance antimagic labelings of G. In this paper we obtain the local distance antimagic labelings for several families of graphs including the path Pn , the cycle Cn , the wheel graph Wn , friendship graph Fn , the corona product of graphs G ° K m ¯ , complete multipartite graph and some special types of the caterpillars. We also find upper bounds for the local distance antimagic chromatic number for these families of graphs.

On the first outdegree Zagreb index of a digraph

Let D be a digraph of order n and having a arcs. Let d1+,d2+,…,dn+ be the vertex outdegrees of D. The first Zagreb index of D is denoted by Zg+(D) and is defined as Zg+(D)=∑i=1ndi+. We obtain several upper and lower bounds for the first outdegree Zagreb index Zg+(D) of a digraph D in terms of the number of vertices n and the number of arcs a. We characterize the extremal digraphs attaining these bounds. Further, we determine the digraphs which attain the maximum, the second maximum and the third maximum value for Zg+(D) among all digraphs of order n and among all bipartite digraphs of order n. We determine the digraphs which attain the maximum value for Zg+(D) among all digraphs with its underlying graph having m≤n+2 edges. Also, we discuss the orientations of a path Pn and show that the path with n−1 vertices of outdegree 1 attains the minimum value for ZG+(D).

BILANGAN KETERHUBUNGAN TITIK PELANGI BEBERAPA KELAS GRAF

A graph G is called a rainbow vertex connected if every two vertices G are connected by a rainbow path, that is, a path whose all the internal vertices are of a different color. The rainbow vertex connection number of graph G denoted by rvc(G) is the minimum number of colors used to color all vertices by G such that the graph G is connected to rainbow vertex. The rainbow vertex connection number in a graph will not be less than the diameter of the graph minus one. The rainbow vertex connection number discussed in this article for various classes of graphs include complete graph Kn, complete bipartite graph Km,n , wheel graph Wn , two-layer wheel graph Wn2, complete multipartite graph Kn1,n2,...,nt , path Pn, comb graph GSn, graph , graph , graph , graph .
 Keywords: graph, vertex coloring, rainbow vertex connection number.

Optimal Fault-Tolerant Resolving Set of Power Paths

In a simple connected undirected graph G, an ordered set R of vertices is called a resolving set if for every pair of distinct vertices u and v, there is a vertex w∈R such that d(u,w)≠d(v,w). A resolving set F for the graph G is a fault-tolerant resolving set if for each v∈F, F∖{v} is also a resolving set for G. In this article, we determine an optimal fault-resolving set of r-th power of any path Pn when n≥r(r−1)+2. For the other values of n, we give bounds for the size of an optimal fault-resolving set. We have also presented an algorithm to construct a fault-tolerant resolving set of Pmr from a fault-tolerant resolving set of Pnr where m<n.

K]-Roman Domination in Digraphs

Let D=(V(D),A(D)) be a finite, simple digraph and k a positive integer. A function f:V(D)→{0,1,2,…,k+1} is called a [k]-Roman dominating function (for short, [k]-RDF) if f(AN−[v])≥|AN−(v)|+k for any vertex v∈V(D), where AN−(v)={u∈N−(v):f(u)≥1} and AN−[v]=AN−(v)∪{v}. The weight of a [k]-RDF f is ω(f)=∑v∈V(D)f(v). The minimum weight of any [k]-RDF on D is the [k]-Roman domination number, denoted by γ[kR](D). For k=2 and k=3, we call them the double Roman domination number and the triple Roman domination number, respectively. In this paper, we presented some general bounds and the Nordhaus–Gaddum bound on the [k]-Roman domination number and we also determined the bounds on the [k]-Roman domination number related to other domination parameters, such as domination number and signed domination number. Additionally, we give the exact values of γ[kR](Pn) and γ[kR](Cn) for the directed path Pn and directed cycle Cn.

Bounds on radio mean number of graphs

A radio mean labeling l of G maps distinct vertices of G to distinct elements of ℤ + satisfying the radio mean condition that diam ( G ) + 1 - d G ( w , w ′ ) ≤ ⌈ l ( w ) + l ( w ′ ) 2 ⌉ , ∀ w , w ′ ∈ V ( G ) where dG (w, w′) is the smallest length of a w, w′- path in G and diam (G) = max {dG (w, w′) : w, w′ ∈ V (G)} is the diameter of G. The radio mean number of graph G is defined as rmn (G) = min {span (l) : l isaradiomeanlabelingof G} where span(l) is given by max {l (w) : w ∈ V (G)}. It has been proved in literature that |V (G) | ≤ rmn (G) ≤ |V (G) | + diam (G) -2. Cryptographic algorithms can exploit the unique radio mean number associated with a graph to generate keys. An exhaustive listing of all feasible radio mean labelings and their span is essential to obtain the radio mean number of a given graph. Since the radio mean condition depends on the distance between vertices and the graph’s diameter, as the order and diameter increase, finding a radio mean labeling itself is quite difficult and, so is obtaining the radio mean number of a given graph. Here we discuss the extreme values of the radio mean number of a given graph of order n. In this article we obtained bounds on the radio mean number of a graph G of order n and diameter d in terms of the radio mean number of its induced subgraph H where diam (H) = d and dH (w, w′) = dG (w, w′) for any w, w′ ∈ V (H). The diametral path Pd+1 is one such induced subgraph of G and hence we have deduced the limits of rmn (G) in terms of rmn (Pd+1). It is known that if d = 1, 2 or 3, then rmn (G) = n. Here, we have given alternative proof for the same. The authors of this article have studied radio mean labeling of paths in another article. Using those results, we have improved the bounds on the radio mean number of a graph of order n and diameter d ≥ 4. It is also shown that among all connected graphs on n vertices, the path Pn of order n possesses the maximum radio mean number. This is the first article that has completely solved the question of maximum and minimum attainable radio mean numbers of graphs of order n.

Pebbling in powers of paths

The t-fold pebbling number, πt(G), of a graph G is defined to be the minimum number m so that, from any given configuration of m pebbles on the vertices of G, it is possible to place at least t pebbles on any specified vertex via pebbling moves. It has been conjectured that the pebbling numbers of pyramid-free chordal graphs can be calculated in polynomial time.The kth power G(k) of the graph G is obtained from G by adding an edge between any two vertices of distance at most k from each other. The kth power of the path Pn on n vertices is an important class of pyramid-free chordal graphs, and is a stepping stone to the more general class of k-paths and the still more general class of interval graphs. Pachter, Snevily, and Voxman (1995) calculated π(Pn(2)), Kim (2004) calculated π(Pn(3)), and Kim and Kim (2010) calculated π(Pn(4)). In this paper we calculate πt(Pn(k)) for all n, k, and t.For a function D:V(G)→N, the D-pebbling number, π(G,D), of a graph G is defined to be the minimum number m so that, from any given configuration of m pebbles on the vertices of G, it is possible to place at least D(v) pebbles on each vertex v via pebbling moves. It has been conjectured that π(G,D)≤π|D|(G) for all G and D. We make the stronger conjecture that every G and D satisfies π(G,D)≤π|D|(G)−(s(D)−1), where s(D) counts the number of vertices v with D(v)>0. We prove that trees and Pn(k), for all n and k, satisfy the stronger conjecture.The pebbling exponenteπ(G) of a graph G was defined by Pachter et al., to be the minimum k for which π(G(k))=n(G(k)). Of course, eπ(G)≤diam(G), and Czygrinow, Hurlbert, Kierstead, and Trotter (2002) proved that almost all graphs G have eπ(G)=1. Lourdusamy and Mathivanan (2015) proved several results on πt(Cn2), and Hurlbert (2017) proved an asymptotically tight formula for eπ(Cn). Our formula for πt(Pn(k)) allows us to compute eπ(Pn) within additively narrow bounds.

Disproof of a conjecture on the minimum Wiener index of signed trees

The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices. Sam Spiro [The Wiener index of signed graphs, Appl. Math. Comput., 416(2022)126755] recently introduced the Wiener index for a signed graph and conjectured that the path Pn with alternating signs has the minimum Wiener index among all signed trees with n vertices. By constructing an infinite family of counterexamples, we prove that the conjecture is false whenever n is at least 30.

Decomposition dimension of corona product of some classes of graphs

For an ordered k-decomposition D = {G1, G2,...,Gk} of a connected graph G = (V,E), the D-representation of an edge e is the k-tuple γ(e/D)=(d(e, G1), d(e, G2), ...,d(e, Gk)), where d(e, Gi) represents the distance from e to Gi. A decomposition D is resolving if every two distinct edges of G have distinct representations. The minimum k for which G has a resolving k-decomposition is its decomposition dimension dec(G). In this paper, the decomposition dimension of corona product of the path Pn and cycle Cn with the complete graphs K1 and K2 are determined.

Secure Domination Cover Pebbling Number of Join of graphs

Objectives: To find the secure domination cover pebbling number for the join of two graphs G(p; q) and G ′ (p ′ ;q ′ ). Methods: We define Secure domination cover pebbling number, fsd p(G), of a graph G as the minimum number of pebbles that must be placed on V(G) such that, after a sequence of pebbling moves, the set of vertices with pebbles forms a secure dominating set for G. Findings: We found the secure domination cover pebbling number for the join of two graphs G(p; q) and Kn. Also, the secure domination cover pebbling number for the join of two graphs G(p; q) and G ′ (p ′ ;q ′ ) is determined when the cardinality of the secure dominating set is 2, 3 and 4. A generalization for the secure domination cover pebbling number of path Pn is also found. Subject Mathematics Classification: 05C38, 05C69 Keywords: Graph pebbling; Secure domination; Cover pebbling number; Secure domination cover pebbling number 2010 Subject Mathematics Classification: 05C38; 05C69

A REMARK ON THE EDGE IRREGULARITY STRENGTH OF CORONA PRODUCT OF TWO PATHS

With respect to a simple graph G, a vertex labeling ϕ: V(G) > {1,2,...,k) is known as k-labeling. The weight corresponding to an edge xy in G, expressed as wϕ (xy), represents the labels sum of end vertices x and y, given by wϕ (xy) = ϕ(x) + ϕ(y) A vertex k-labeling is expressed as an edge irregular k-labeling with respect to graph G provided that for every two distinct edges e and f, there exists wϕ(e) ≠ wϕ(f) Here, the minimum k where the graph G possesses an edge irregular k-labeling is known as the edge irregularity strength with respect to G, expressed as (G). Here, we examine the edge irregularity strength’s exact value of corona product with respect to two paths Pn and Pm , in which n ≥ 2 and m = 3, 4, 5.

Linear Forest mP3 Plus a Longer Path Pn Becoming Antimagic

An edge labeling of a graph G is a bijection f from E(G) to a set of |E(G)| integers. For a vertex u in G, the induced vertex sum of u, denoted by f+(u), is defined as f+(u)=∑uv∈E(G)f(uv). Graph G is said to be antimagic if it has an edge labeling g such that g(E(G))={1,2,⋯,|E(G)|} and g+(u)≠g+(v) for any pair u,v∈V(G) with u≠v. A linear forest is a union of disjoint paths of orders greater than one. Let mPk denote a linear forest consisting of m disjoint copies of path Pk. It is known that mP3 is antimagic if and only if m=1. In this study, we add a disjoint path Pn (n≥4) to mP3 and develop a necessary condition and a sufficient condition whereby the new linear forest mP3⋃Pn may be antimagic.

Parity Properties of Configurations

In the paper, the crossing number of the join product G*+Dn for the disconnected graph G* consisting of two components isomorphic to K2 and K3 is given, where Dn consists of n isolated vertices. Presented proofs are completed with the help of the graph of configurations that is a graphical representation of minimum numbers of crossings between two different subgraphs whose edges do not cross the edges of G*. For the first time, multiple symmetry between configurations are presented as parity properties. We also determine crossing numbers of join products of G* with paths Pn and cycles Cn on n vertices by adding new edges joining vertices of Dn.

The access time of random walks on trees with given partition

Denote by T(s,t) the set of trees, whose vertex set can be partitioned into two independent sets of sizes s and t respectively. Given a tree T with stationary distribution π and a vertex v∈T, the access time HT(π,v) is the expected length of optimal stopping rules from π to v. In this paper, we get a sharp upper bound for maxv∈THT(π,v) and a sharp lower bound for minv∈THT(π,v) among T(s,t), respectively. The corresponding extremal graphs are also obtained. As a byproduct, it is proved that the path Pn maximizes maxv∈THT(π,v) and the star K1,n−1 minimizes minv∈THT(π,v) among all trees on n vertices.

Pancyclicity of the n-Generalized Prism over Skirted Graphs

A side skirt is a planar rooted tree T, T≠P2, where the root of T is a vertex of degree at least two, and all other vertices except the leaves are of degree at least three. A reduced Halin graph or a skirted graph is a plane graph G=T∪P, where T is a side skirt, and P is a path connecting the leaves of T in the order determined by the embedding of T. The structure of reduced Halin or skirted graphs contains both symmetry and asymmetry. For n≥2 and Pn=v1v2v3⋯vn as a path of length n−1, we call the Cartesian product of a graph G and a path Pn, the n-generalized prism over a graph G. We have known that the n-generalized prism over a skirted graph is Hamiltonian. To support the Bondy’s metaconjecture from 1971, we show that the n-generalized prism over a skirted graph is pancyclic.

Vertex pancyclicity over lexicographic products

A lexicographic product of graphs G and H, denoted by is defined as a graph with the vertex set and an edge presents in the product whenever or (u 1 = u 2 and ). We investigate the sufficient conditions for vertex pancyclicity of lexicographic products of complete graphs Kn , paths Pn or cycles Cn with a general graph. We obtain that (i) if G 1 is a traceable graph of even order and G 2 is a graph with at least one edge, then is vertex pancyclic; (ii) if G 1 is hamiltonian and G 2 is a graph with at least one edge, then is vertex pancyclic.

Even -odd average harmonious graph

This article studies we have presented an innovative harmonious labelling is Even odd average harmonious graph labeling. G is an Even odd average harmonic graph because it contains n vertices and m edges if such a function exists f:V(G)→{1,3,5,⋯(2n-1)} as the induced mapping f∗:E(G)→0,1,2,⋯.m-1} characterized as f∗(uv) = fu+f(v)2 (modm) is a bijection. The function fis said to be an Even the odd average harmonious graph G. A graph which admit an Even the odd average harmonious graph. We demonstrated in this research that the graph path Pn, cycle Cn, the star graph Sq+1, the graph Pn2, the bistar graph Bp,q, the comb Pr⊙K1, the graph C3@pK1, the graph C2r-1@K1 are even odd Average Harmonious Labeling of graphs.