$$Z_3$$ Z 3 -Connectivity with Independent Number 2

2015 ◽  
Vol 32 (1) ◽  
pp. 419-429 ◽  
Author(s):  
Fan Yang ◽  
Xiangwen Li ◽  
Liangchen Li
Keyword(s):  
Independent Number

A Note of Independent Number and Domination Number of Qn,k,m-Graph

2019 ◽  
Vol 29 (03) ◽  
pp. 1950011
Author(s):  
Jiafei Liu ◽  
Shuming Zhou ◽  
Zhendong Gu ◽  
Yihong Wang ◽  
Qianru Zhou
Keyword(s):  
Interconnection Networks ◽  
Interconnection Network ◽  
Domination Number ◽  
Multiprocessor System ◽  
Star Graph ◽  
Minimum Cardinality ◽  
Arrangement Graph ◽  
Independent Number ◽  
Potential Applications ◽  
By Products

The independent number and domination number are two essential parameters to assess the resilience of the interconnection network of multiprocessor systems which is usually modeled by a graph. The independent number, denoted by [Formula: see text], of a graph [Formula: see text] is the maximum cardinality of any subset [Formula: see text] such that no two elements in [Formula: see text] are adjacent in [Formula: see text]. The domination number, denoted by [Formula: see text], of a graph [Formula: see text] is the minimum cardinality of any subset [Formula: see text] such that every vertex in [Formula: see text] is either in [Formula: see text] or adjacent to an element of [Formula: see text]. But so far, determining the independent number and domination number of a graph is still an NPC problem. Therefore, it is of utmost importance to determine the number of independent and domination number of some special networks with potential applications in multiprocessor system. In this paper, we firstly resolve the exact values of independent number and upper and lower bound of domination number of the [Formula: see text]-graph, a common generalization of various popular interconnection networks. Besides, as by-products, we derive the independent number and domination number of [Formula: see text]-star graph [Formula: see text], [Formula: see text]-arrangement graph [Formula: see text], as well as three special graphs.

scholarly journals Semi Square Stable Graphs

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 597
Author(s):  
Mohammad Abudayah ◽  
Omar Alomari
Keyword(s):  
Independent Set ◽  
Interval Graphs ◽  
Maximum Independent Set ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
New Class ◽  
Dominating Number ◽  
Independent Number ◽  
Necessary And Sufficient

The independent number of a graph G is the cardinality of the maximum independent set of G, denoted by α ( G ) . The independent dominating number is the cardinality of the smallest independent set that dominates all vertices of G. In this paper, we introduce a new class of graphs called semi-square stable for which α ( G 2 ) = i ( G ) . We give a necessary and sufficient condition for a graph to be semi-square stable, and we study when interval graphs are semi-square stable.

scholarly journals On the relations between arboricity and independent number or covering number

1998 ◽  
Vol 87 (1-3) ◽  
pp. 279-283
Author(s):  
Zhang Zhongfu ◽  
Liu Linzhong ◽  
Zhang Jianxun ◽  
Wang Jianfang
Keyword(s):  
Covering Number ◽  
Independent Number

Second-order piezomagnetism in polychromatic crystals

1990 ◽  
Vol 46 (2) ◽  
pp. 130-133
Author(s):  
K. Rama Mohana Rao
Keyword(s):  
Point Group ◽  
Physical Property ◽  
Theoretical Method ◽  
Second Order ◽  
Acta Cryst ◽  
Theoretical Methods ◽  
Group Theoretical Method ◽  
Group 4 ◽  
Independent Number

The group-theoretical method established for obtaining the non-vanishing independent number of constants required to describe a magnetic/physical property in respect of the 18 polychromatic crystal classes [Rama Mohana Rao (1987). J. Phys. A, 20, 47-57] has been explored to enumerate the second- order piezomagnetic coefficients (n i ′) for the same classes. The advantage of Jahn's method [Jahn (1949). Acta Cryst. 2, 30-33] is appreciated in obtaining these n i ′ through the reduction of a representation. The different group-theoretical methods are illustrated with the help of the point group 4. The results obtained for all 18 classes are tabulated and briefly discussed.

BOUNDS ON HARARY INDEX WITH RESPECT TO VERTEX CONNECTIVITY, INDEPENDENT NUMBER AND INDEPENDENT DOMINATION NUMBER OF A GRAPH

2020 ◽  
Vol 9 (11) ◽  
pp. 9319-9328
Author(s):  
V.V. Manjalapur ◽  
M.B. Rotti
Keyword(s):  
Domination Number ◽  
Molecular Graph ◽  
Vertex Connectivity ◽  
Harary Index ◽  
Independent Number ◽  
Independent Domination

In the present paper, we obtain bounds for Harary index $H(G)$ of a connected (molecular) graph in terms of vertex connectivity, independent number, independent domination number and characterize graphs extremal with respect to them.

Total graph of the ring ℤn × ℤm

2015 ◽  
Vol 07 (01) ◽  
pp. 1550004 ◽  
Author(s):  
Alpesh M. Dhorajia
Keyword(s):  
Commutative Ring ◽  
Undirected Graph ◽  
Clique Number ◽  
Total Graph ◽  
Fundamental Properties ◽  
Zero Divisors ◽  
Independent Number ◽  
Positive Integers

Let R be a commutative ring and Z(R) be the set of all zero-divisors of R. The total graph of R, denoted by T Γ(R), is the (undirected) graph with vertices set R. For any two distinct elements x, y ∈ R, the vertices x and y are adjacent if and only if x + y ∈ Z(R). In this paper, we obtain certain fundamental properties of the total graph of ℤn × ℤm, where n and m are positive integers. We determine the clique number and independent number of the total graph T Γ(ℤn × ℤm).

scholarly journals On the dominating number, independent number and the regularity of the relative co-prime graph of a group

2017 ◽  
Vol 13 (2) ◽  
Author(s):  
Norarida Abd Rhani ◽  
Nor Muhainiah Mohd Ali ◽  
Nor Haniza Sarmin ◽  
Ahmad Erfanian
Keyword(s):  
Finite Group ◽  
Dominating Set ◽  
Prime Graph ◽  
Greatest Common Divisor ◽  
Dominating Number ◽  
Independent Number ◽  
A Finite Group

Let H be a subgroup of a finite group G. The co-prime graph of a group is defined as a graph whose vertices are elements of G and two distinct vertices are adjacent if and only if the greatest common divisor of order of x and y is equal to one. This concept has been extended to the relative co-prime graph of a group with respect to a subgroup H, which is defined as a graph whose vertices are elements of G and two distinct vertices x and y are joined by an edge if and only if their orders are co-prime and any of x or y is in H.  Some properties of graph such as the dominating number, degree of a dominating set of order one and independent number are obtained. Lastly, the regularity of the relative co-prime graph of a group is found.

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