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

scholarly journals On r-dynamic coloring of comb graphs

2021 ◽  
Vol 27 (2) ◽  
pp. 191-200
Author(s):  
K. Kalaiselvi ◽  
◽  
N. Mohanapriya ◽  
J. Vernold Vivin ◽  
◽  
...  
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Upper Bound ◽  
Line Graph ◽  
Proper Coloring ◽  
Total Graph ◽  
Middle Graph ◽  
Different Color

An r-dynamic coloring of a graph G is a proper coloring of G such that every vertex in V(G) has neighbors in at least $\min\{d(v),r\}$ different color classes. The r-dynamic chromatic number of graph G denoted as $\chi_r (G)$, is the least k such that G has a coloring. In this paper we obtain the r-dynamic chromatic number of the central graph, middle graph, total graph, line graph, para-line graph and sub-division graph of the comb graph $P_n\odot K_1$ denoted by $C(P_n\odot K_1), M(P_n\odot K_1), T(P_n\odot K_1), L(P_n\odot K_1), P(P_n\odot K_1)$ and $S(P_n\odot K_1)$ respectively by finding the upper bound and lower bound for the r-dynamic chromatic number of the Comb graph.

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.

Cyclic Stress Relaxation (Csr) of Filled Rubber and Rubber Components

2009 ◽  
Vol 82 (1) ◽  
pp. 104-112 ◽  
Author(s):  
S. Asare ◽  
A. G. Thomas ◽  
J. J. C. Busfield
Keyword(s):  
Stress Relaxation ◽  
Line Graph ◽  
Cyclic Stress ◽  
Maximum Displacement ◽  
Real Component ◽  
Test Pieces ◽  
Filled Rubber ◽  
Straight Line ◽  
Relaxation Measurements ◽  
Number Of Cycles

Abstract Under repeated stressing it is well known that rubber materials exhibit cyclic stress relaxation (CSR). Previous work has shown that the amount of relaxation observed from cycle to cycle is significantly greater than that expected from static relaxation measurements. The reduction in the stress attained on the second and successive loading cycles as compared to the stress attained on the first cycle in a stress strain cyclic test of fixed amplitude has been measured for elastomer test pieces and engineering components. It is seen that the peak force, under cyclic testing to a specific maximum displacement, plotted against the number of cycles on logarithmic scales produces a straight line graph, whose slope correlates to the rate of cyclic stress relaxation per decade. The rate of cyclic stress relaxation was found to increase with displacement amplitude in all modes of deformation. Plotting the rate of stress relaxation per decade against the maximum average strain energy attained in the cycle reduces the data measured in different deformation modes for both simple test pieces and components to a single curve. This approach allows the cyclic stress relaxation in a real component to be predicted from simple laboratory tests.

scholarly journals On the nullity of connected graphs with least eigenvalue at least -2

2013 ◽  
Vol 7 (2) ◽  
pp. 250-261 ◽  
Author(s):  
Jiang Zhou ◽  
Lizhu Sun ◽  
Hongmei Yao ◽  
Changjiang Bu
Keyword(s):  
Upper Bound ◽  
Adjacency Matrix ◽  
Line Graph ◽  
Line Graphs ◽  
Connected Graphs ◽  
Least Eigenvalue

Let L (resp. L+) be the set of connected graphs with least adjacency eigenvalue at least -2 (resp. larger than -2). The nullity of a graph G, denoted by ?(G), is the multiplicity of zero as an eigenvalue of the adjacency matrix of G. In this paper, we give the nullity set of L+ and an upper bound on the nullity of exceptional graphs. An expression for the nullity of generalized line graphs is given. For G ? L, if ?(G) is sufficiently large, then G is a proper generalized line graph (G is not a line graph).

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 .

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 An upper bound for the chromatic number of line graphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Andrew D. King ◽  
Bruce A. Reed ◽  
Adrian R. Vetta
Keyword(s):  
Chromatic Number ◽  
Polynomial Time ◽  
Upper Bound ◽  
Maximum Degree ◽  
Line Graph ◽  
Polynomial Time Algorithm ◽  
Clique Number ◽  
Time Algorithm ◽  
Line Graphs ◽  
International Audience

International audience It was conjectured by Reed [reed98conjecture] that for any graph $G$, the graph's chromatic number $χ (G)$ is bounded above by $\lceil Δ (G) +1 + ω (G) / 2\rceil$ , where $Δ (G)$ and $ω (G)$ are the maximum degree and clique number of $G$, respectively. In this paper we prove that this bound holds if $G$ is the line graph of a multigraph. The proof yields a polynomial time algorithm that takes a line graph $G$ and produces a colouring that achieves our bound.

scholarly journals Bounds for Incidence Energy of Some Graphs

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Weizhong Wang ◽  
Dong Yang
Keyword(s):  
Regular Graph ◽  
Upper Bound ◽  
Maximum Degree ◽  
Incidence Matrix ◽  
Line Graph ◽  
Singular Values ◽  
Simple Graph

LetGbe a simple graph. The incidence energy (IEfor short) ofGis defined as the sum of the singular values of the incidence matrix. In this paper, a new upper bound forIEof graphs in terms of the maximum degree is given. Meanwhile, bounds forIEof the line graph of a semiregular graph and the paraline graph of a regular graph are obtained.

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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