scholarly journals The Optimal Rubbling Number of Paths, Cycles, and Grids

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Zheng-Jiang Xia ◽  
Zhen-Mu Hong
Keyword(s):  
Upper Bound ◽  
Adjacent Vertex ◽  
Number Of Cycles

A pebbling move on a graph G consists of the removal of two pebbles from one vertex and the placement of one pebble on an adjacent vertex. Rubbling is a version of pebbling where an additional move is allowed, which is also called the strict rubbling move. In this new move, one pebble each is removed from u and v adjacent to a vertex w , and one pebble is added on w . The rubbling number of a graph G is the smallest number m , such that one pebble can be moved to each vertex from every distribution with m pebbles. The optimal rubbling number of a graph G is the smallest number m , such that one pebble can be moved to each vertex from some distribution with m pebbles. In this paper, we give short proofs to determine the rubbling number of cycles and the optimal rubbling number of paths, cycles, and the grid P 2 × P n ; moreover, we give an upper bound of the optimal rubbling number of P m × P n .

scholarly journals Edge Mostar Indices of Cacti Graph With Fixed Cycles

2021 ◽  
Vol 9 ◽  
Author(s):  
Farhana Yasmeen ◽  
Shehnaz Akhter ◽  
Kashif Ali ◽  
Syed Tahir Raza Rizvi
Keyword(s):  
Upper Bound ◽  
Connected Graph ◽  
Thermodynamic Characteristics ◽  
Chemical Compounds ◽  
Fixed Number ◽  
Common Vertex ◽  
Topological Invariants ◽  
Cactus Graph ◽  
Number Of Cycles

Topological invariants are the significant invariants that are used to study the physicochemical and thermodynamic characteristics of chemical compounds. Recently, a new bond additive invariant named the Mostar invariant has been introduced. For any connected graph ℋ, the edge Mostar invariant is described as Moe(ℋ)=∑gx∈E(ℋ)|mℋ(g)−mℋ(x)|, where mℋ(g)(or mℋ(x)) is the number of edges of ℋ lying closer to vertex g (or x) than to vertex x (or g). A graph having at most one common vertex between any two cycles is called a cactus graph. In this study, we compute the greatest edge Mostar invariant for cacti graphs with a fixed number of cycles and n vertices. Moreover, we calculate the sharp upper bound of the edge Mostar invariant for cacti graphs in ℭ(n,s), where s is the number of cycles.

scholarly journals Locating-Total Domination Number of Cacti Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Jianxin Wei ◽  
Uzma Ahmad ◽  
Saira Hameed ◽  
Javaria Hanif
Keyword(s):  
Upper Bound ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Total Domination Number ◽  
Unicyclic Graphs ◽  
Total Dominating Set ◽  
Bicyclic Graphs ◽  
Number Of Cycles

For a connected graph J, a subset W ⊆ V J is termed as a locating-total dominating set if for a ∈ V J ,   N a ∩ W ≠ ϕ , and for a ,   b ∈ V J − W ,   N a ∩ W ≠ N b ∩ W . The number of elements in a smallest such subset is termed as the locating-total domination number of J. In this paper, the locating-total domination number of unicyclic graphs and bicyclic graphs are studied and their bounds are presented. Then, by using these bounds, an upper bound for cacti graphs in terms of their order and number of cycles is estimated. Moreover, the exact values of this domination variant for some families of cacti graphs including tadpole graphs and rooted products are also determined.

scholarly journals An improved upper bound on the adjacent vertex distinguishing chromatic index of a graph

2014 ◽  
Vol 162 ◽  
pp. 348-354 ◽  
Author(s):  
Lianzhu Zhang ◽  
Weifan Wang ◽  
Ko-Wei Lih
Keyword(s):  
Upper Bound ◽  
Adjacent Vertex ◽  
Chromatic Index

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].

The upper bound of the number of cycles in a 2-factor of a line graph

2007 ◽  
Vol 55 (1) ◽  
pp. 72-82 ◽  
Author(s):  
Jun Fujisawa ◽  
Liming Xiong ◽  
Kiyoshi Yoshimoto ◽  
Shenggui Zhang
Keyword(s):  
Upper Bound ◽  
Line Graph ◽  
Number Of Cycles

On the number of cycles in a graph

2018 ◽  
Vol 68 (1) ◽  
pp. 1-10
Author(s):  
Bader F. AlBdaiwi
Keyword(s):  
Upper Bound ◽  
Hamiltonian Graph ◽  
Cubic Graphs ◽  
Proof Technique ◽  
Number Of Cycles ◽  
Special Classes

AbstractThere is a sizable literature on investigating the minimum and maximum numbers of cycles in a class of graphs. However, the answer is known only for special classes. This paper presents a result on the smallest number of cycles in Hamiltonian 3-connected cubic graphs. Further, it describes a proof technique that could improve an upper bound of the largest number of cycles in a Hamiltonian graph.

scholarly journals The Maximum Number of Cycles in a Graph with Fixed Number of Edges

10.37236/8747 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Andrii Arman ◽  
Sergei Tsaturian
Keyword(s):  
Upper Bound ◽  
Fixed Number ◽  
Number Of Cycles

The main problem considered in this paper is maximizing the number of cycles in a graph with given number of edges. In 2009, Király conjectured that there is constant $c$ such that any graph with $m$ edges has at most $c(1.4)^m$ cycles. In this paper, it is shown that for sufficiently large $m$, a graph with $m$ edges has at most $(1.443)^m$ cycles. For sufficiently large $m$, examples of a graph with $m$ edges and $(1.37)^m$ cycles are presented. For a graph with given number of vertices and edges an upper bound on the maximal number of cycles is given. Also, bounds tight up to a constant are presented for the maximum number of cycles in a multigraph with given number of edges, as well as in a multigraph with given number of vertices and edges.

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