scholarly journals The 2-Pebbling Property of the Middle Graph of Fan Graphs

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Yongsheng Ye ◽  
Fang Liu ◽  
Caixia Shi
Keyword(s):  
Connected Graph ◽  
Adjacent Vertex ◽  
Arbitrary Vertex ◽  
Pebbling Number ◽  
Middle Graph

A pebbling move on a graphGconsists of taking two pebbles off one vertex and placing one pebble on an adjacent vertex. The pebbling number of a connected graphG, denoted byf(G), is the leastnsuch that any distribution ofnpebbles onGallows one pebble to be moved to any specified but arbitrary vertex by a sequence of pebbling moves. This paper determines the pebbling numbers and the 2-pebbling property of the middle graph of fan graphs.

A note on pebbling bound for a product of Class 0 graphs

2021 ◽  
pp. 2250031
Author(s):  
Nopparat Pleanmani ◽  
Somnuek Worawiset
Keyword(s):  
Upper Bound ◽  
Connected Graph ◽  
Adjacent Vertex ◽  
Arbitrary Vertex ◽  
Pebbling Number

Let [Formula: see text] be a connected graph. For a configuration of pebbles on the vertices of [Formula: see text], a pebbling move on [Formula: see text] is the process of taking two pebbles from a vertex and adding one of them on an adjacent vertex. The pebbling number of [Formula: see text], denoted by [Formula: see text], is the least number of pebbles to guarantee that for any configuration of pebbles on [Formula: see text] and arbitrary vertex [Formula: see text], there is a sequence of pebbling movement that places at least one pebble on [Formula: see text]. The graph [Formula: see text] is said to be of Class 0 if its pebbling number equals its order. For a Class [Formula: see text] connected graph [Formula: see text], we improve a recent upper bound for [Formula: see text] in terms of [Formula: see text].

Graham’s pebbling conjecture holds for the product of a graph and a sufficiently large complete bipartite graph

2019 ◽  
Vol 11 (06) ◽  
pp. 1950068
Author(s):  
Nopparat Pleanmani
Keyword(s):  
Bipartite Graph ◽  
Network Optimization ◽  
Connected Graph ◽  
Complete Bipartite Graph ◽  
Fixed Number ◽  
Arbitrary Vertex ◽  
Graph Pebbling ◽  
Vertex Set ◽  
Graham’S Conjecture ◽  
Pebbling Number

A graph pebbling is a network optimization model for the transmission of consumable resources. A pebbling move on a connected graph [Formula: see text] is the process of removing two pebbles from a vertex and placing one of them on an adjacent vertex after configuration of a fixed number of pebbles on the vertex set of [Formula: see text]. The pebbling number of [Formula: see text], denoted by [Formula: see text], is defined to be the least number of pebbles to guarantee that for any configuration of pebbles on [Formula: see text] and arbitrary vertex [Formula: see text], there is a sequence of pebbling movement that places at least one pebble on [Formula: see text]. For connected graphs [Formula: see text] and [Formula: see text], Graham’s conjecture asserted that [Formula: see text]. In this paper, we show that such conjecture holds when [Formula: see text] is a complete bipartite graph with sufficiently large order in terms of [Formula: see text] and the order of [Formula: see text].

Herscovici’s conjecture on C2n × G

2020 ◽  
Vol 12 (05) ◽  
pp. 2050071
Author(s):  
A. Lourdusamy ◽  
T. Mathivanan
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Adjacent Vertex ◽  
Connected Graphs ◽  
Number Formula ◽  
Pebbling Number

The [Formula: see text]-pebbling number, [Formula: see text], of a connected graph [Formula: see text], is the smallest positive integer such that from every placement of [Formula: see text] pebbles, [Formula: see text] pebbles can be moved to any specified target vertex by a sequence of pebbling moves, each move taking two pebbles off a vertex and placing one on an adjacent vertex. A graph [Formula: see text] satisfies the [Formula: see text]-pebbling property if [Formula: see text] pebbles can be moved to any specified vertex when the total starting number of pebbles is [Formula: see text], where [Formula: see text] is the number of vertices with at least one pebble. We show that the cycle [Formula: see text] satisfies the [Formula: see text]-pebbling property. Herscovici conjectured that for any connected graphs [Formula: see text] and [Formula: see text], [Formula: see text]. We prove Herscovici’s conjecture is true, when [Formula: see text] is an even cycle and for variety of graphs [Formula: see text] which satisfy the [Formula: see text]-pebbling property.

scholarly journals The 2-pebbling property of squares of paths and Graham’s conjecture

2020 ◽  
Vol 18 (1) ◽  
pp. 87-92
Author(s):  
Yueqing Li ◽  
Yongsheng Ye
Keyword(s):  
Connected Graph ◽  
Adjacent Vertex ◽  
Graham’S Conjecture ◽  
Pebbling Number

Abstract A pebbling move on a graph G consists of taking two pebbles off one vertex and placing one pebble on an adjacent vertex. The pebbling number of a connected graph G, denoted by f(G), is the least n such that any distribution of n pebbles on G allows one pebble to be moved to any specified vertex by a sequence of pebbling moves. In this paper, we determine the 2-pebbling property of squares of paths and Graham’s conjecture on $\begin{array}{} P_{2n}^2 \end{array} $.

scholarly journals The optimal pebbling of spindle graphs

2019 ◽  
Vol 17 (1) ◽  
pp. 582-587
Author(s):  
Ze-Tu Gao ◽  
Jian-Hua Yin
Keyword(s):  
Connected Graph ◽  
Adjacent Vertex ◽  
Left Endpoint ◽  
Pebbling Number ◽  
The Right ◽  
Optimal Pebbling

Abstract Given a distribution of pebbles on the vertices of a connected graph G, a pebbling move on G consists of taking two pebbles off one vertex and placing one on an adjacent vertex. The optimal pebbling number of G, denoted by πopt(G), is the smallest number m such that for some distribution of m pebbles on G, one pebble can be moved to any vertex of G by a sequence of pebbling moves. Let Pk be the path on k vertices. Snevily defined the n–k spindle graph as follows: take n copies of Pk and two extra vertices x and y, and then join the left endpoint (respectively, the right endpoint) of each Pk to x (respectively, y), the resulting graph is denoted by S(n, k), and called the n–k spindle graph. In this paper, we determine the optimal pebbling number for spindle graphs.

scholarly journals The $t$-Pebbling Number is Eventually Linear in $t$

10.37236/640 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Michael Hoffmann ◽  
Jiří Matoušek ◽  
Yoshio Okamoto ◽  
Philipp Zumstein
Keyword(s):  
Initial Distribution ◽  
Adjacent Vertex ◽  
Weighted Graphs ◽  
Graph Pebbling ◽  
Pebbling Number ◽  
Integer Weight

In graph pebbling games, one considers a distribution of pebbles on the vertices of a graph, and a pebbling move consists of taking two pebbles off one vertex and placing one on an adjacent vertex. The $t$-pebbling number $\pi_t(G)$ of a graph $G$ is the smallest $m$ such that for every initial distribution of $m$ pebbles on $V(G)$ and every target vertex $x$ there exists a sequence of pebbling moves leading to a distibution with at least $t$ pebbles at $x$. Answering a question of Sieben, we show that for every graph $G$, $\pi_t(G)$ is eventually linear in $t$; that is, there are numbers $a,b,t_0$ such that $\pi_t(G)=at+b$ for all $t\ge t_0$. Our result is also valid for weighted graphs, where every edge $e=\{u,v\}$ has some integer weight $\omega(e)\ge 2$, and a pebbling move from $u$ to $v$ removes $\omega(e)$ pebbles at $u$ and adds one pebble to $v$.

scholarly journals Some Vertex/Edge-Degree-Based Topological Indices of r -Apex Trees

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Akbar Ali ◽  
Waqas Iqbal ◽  
Zahid Raza ◽  
Ekram E. Ali ◽  
Jia-Bao Liu ◽  
...  
Keyword(s):  
Graph Theory ◽  
Connected Graph ◽  
Nonnegative Integer ◽  
Symmetric Function ◽  
Topological Indices ◽  
Vertex Degree ◽  
Graph Invariants ◽  
Arbitrary Vertex ◽  
Chemical Graph ◽  
Chemical Graph Theory

In chemical graph theory, graph invariants are usually referred to as topological indices. For a graph G , its vertex-degree-based topological indices of the form BID G = ∑ u v ∈ E G β d u , d v are known as bond incident degree indices, where E G is the edge set of G , d w denotes degree of an arbitrary vertex w of G , and β is a real-valued-symmetric function. Those BID indices for which β can be rewritten as a function of d u + d v − 2 (that is degree of the edge u v ) are known as edge-degree-based BID indices. A connected graph G is said to be r -apex tree if r is the smallest nonnegative integer for which there is a subset R of V G such that R = r and G − R is a tree. In this paper, we address the problem of determining graphs attaining the maximum or minimum value of an arbitrary BID index from the class of all r -apex trees of order n , where r and n are fixed integers satisfying the inequalities n − r ≥ 2 and r ≥ 1 .

scholarly journals On the eccentric connectivity coindex in graphs

2021 ◽  
Vol 7 (1) ◽  
pp. 651-666
Author(s):  
Hongzhuan Wang ◽  
◽  
Xianhao Shi ◽  
Ber-Lin Yu
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Topological Index ◽  
Extremal Problems ◽  
Connectivity Index ◽  
Adjacent Vertex ◽  
Extremal Graph ◽  
Eccentric Connectivity Index ◽  
Unicyclic Graphs

<abstract><p>The well-studied eccentric connectivity index directly consider the contribution of all edges in a graph. By considering the total eccentricity sum of all non-adjacent vertex, Hua et al. proposed a new topological index, namely, eccentric connectivity coindex of a connected graph. The eccentric connectivity coindex of a connected graph $ G $ is defined as</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \overline{\xi}^{c}(G) = \sum\limits_{uv\notin E(G)} (\varepsilon_{G}(u)+\varepsilon_{G}(v)). $\end{document} </tex-math></disp-formula></p> <p>Where $ \varepsilon_{G}(u) $ (resp. $ \varepsilon_{G}(v) $) is the eccentricity of the vertex $ u $ (resp. $ v $). In this paper, some extremal problems on the $ \overline{\xi}^{c} $ of graphs with given parameters are considered. We present the sharp lower bounds on $ \overline{\xi}^{c} $ for general connecteds graphs. We determine the smallest eccentric connectivity coindex of cacti of given order and cycles. Also, we characterize the graph with minimum and maximum eccentric connectivity coindex among all the trees with given order and diameter. Additionally, we determine the smallest eccentric connectivity coindex of unicyclic graphs with given order and diameter and the corresponding extremal graph is characterized as well.</p></abstract>

ON mRJ REACHABILITY IN TREES

2012 ◽  
Vol 04 (04) ◽  
pp. 1250055 ◽  
Author(s):  
BISWAJIT DEB ◽  
KALPESH KAPOOR ◽  
SUKANTA PATI
Keyword(s):  
Connected Graph ◽  
Initial Position ◽  
Adjacent Vertex ◽  
The Other ◽  
Simple Move

Given a tree T, a configuration of T is denoted by [Formula: see text] which represents that there is a robot at the vertex u, a hole at the vertex v and obstacles in the remaining vertices of T. By an mRJ move we mean that the robot is moved from the vertex u to a vertex v having a hole by jumping over m obstacles along a path. The case m = 0 is a simple move of taking the robot from u to the adjacent vertex v with a hole. We investigate the problem of moving a robot from its initial position to all the other vertices using mRJ moves (for some fixed m) in addition to simple moves. A tree is said to be mRJ reachable if there exists a configuration from which it is possible to take the robot to any vertex of the tree using simple or mRJ moves. A connected graph is 1RJ reachable. However, for m ≥ 2 there exists graphs that are not mRJ reachable. We characterize 2RJ and 3RJ reachable trees and give bound for the diameter of mRJ reachable trees.

scholarly journals Strong Menger Connectedness of Augmented k-ary n-cubes

2020 ◽  
Author(s):  
Mei-Mei Gu ◽  
Jou-Ming Chang ◽  
Rong-Xia Hao
Keyword(s):  
Connected Graph ◽  
Fault Tolerant ◽  
Disjoint Paths ◽  
Edge Connectivity ◽  
Arbitrary Vertex ◽  
Edge Disjoint ◽  
Vertex Set ◽  
Delta G ◽  
Disjoint Edge

Abstract A connected graph $G$ is called strongly Menger (edge) connected if for any two distinct vertices $x,y$ of $G$, there are $\min \{\textrm{deg}_G(x), \textrm{deg}_G(y)\}$ internally disjoint (edge disjoint) paths between $x$ and $y$. Motivated by parallel routing in networks with faults, Oh and Chen (resp., Qiao and Yang) proposed the (fault-tolerant) strong Menger (edge) connectivity as follows. A graph $G$ is called $m$-strongly Menger (edge) connected if $G-F$ remains strongly Menger (edge) connected for an arbitrary vertex set $F\subseteq V(G)$ (resp. edge set $F\subseteq E(G)$) with $|F|\leq m$. A graph $G$ is called $m$-conditional strongly Menger (edge) connected if $G-F$ remains strongly Menger (edge) connected for an arbitrary vertex set $F\subseteq V(G)$ (resp. edge set $F\subseteq E(G)$) with $|F|\leq m$ and $\delta (G-F)\geq 2$. In this paper, we consider strong Menger (edge) connectedness of the augmented $k$-ary $n$-cube $AQ_{n,k}$, which is a variant of $k$-ary $n$-cube $Q_n^k$. By exploring the topological proprieties of $AQ_{n,k}$, we show that $AQ_{n,3}$ (resp. $AQ_{n,k}$, $k\geq 4$) is $(4n-9)$-strongly (resp. $(4n-8)$-strongly) Menger connected for $n\geq 4$ (resp. $n\geq 2$) and $AQ_{n,k}$ is $(4n-4)$-strongly Menger edge connected for $n\geq 2$ and $k\geq 3$. Moreover, we obtain that $AQ_{n,k}$ is $(8n-10)$-conditional strongly Menger edge connected for $n\geq 2$ and $k\geq 3$. These results are all optimal in the sense of the maximum number of tolerated vertex (resp. edge) faults.

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