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The vertex (edge) independence number, vertex (edge) cover number and the least eigenvalue of a graph

Ying-Ying Tan, Yi-Zheng Fan
2010 Linear Algebra and its Applications  
among all graphs of a fixed order and a given vertex (edge) independence number or vertex (edge) cover number, and get some bounds for the vertex (edge) independence number, vertex (edge) cover number  ...  Keywords: Graph Adjacency matrix Vertex (edge) independence number Vertex (edge) cover number Least eigenvalue In this paper we characterize the unique graph whose least eigenvalue attains the minimum  ...  The vertex (edge) independence number of G, denoted by α(G) (α (G)), is the maximum of the cardinalities of all vertex (edge) independent sets. • A vertex (edge) cover of a graph G is a set of vertices  ... 
doi:10.1016/j.laa.2010.04.009 fatcat:ymj6l6be7ravrlaryabgdth74a

On intersecting hypergraphs

Barry Guiduli, Zoltán Király
1998 Discrete Mathematics  
We investigate the following question: 'Given an intersecting multi-hypergraph on n points, what fraction of edges must be covered by any of the best 2 points?'  ...  To better study this problem, we introduce the concept of fractional matchings and coverings of order 2.  ...  The set of edges covered by a vertex v is the set {X E E(H): vEX}, and the set of edges covered by a set of vertices is the union of the sets covered by each vertex.  ... 
doi:10.1016/s0012-365x(97)00136-2 fatcat:segaq44abjb65mzojpn4xgafda

Some remarks on normalized matching

Douglas B West, Lawrence H Harper, David E Daykin
1983 Journal of combinatorial theory. Series A  
To obtain (3), we discuss "minimal sets" of covering relations between two adjacent levels of an LYM-order.  ...  Then (XC [n]: ai < lXnA,l <b,}, ordered by inclusion, is a poset having the LYM property. (2) The smallest regular covering of an LYM order has M(P) chains, where M(P) is the least common multiple of the  ...  Similarly, the order on the edges from x induces a set B of partial sums for the flow multiplicites. Note that 1 B 1 is the number of elements covering x.  ... 
doi:10.1016/0097-3165(83)90015-8 fatcat:dkvc62tkc5dnxblxeptopecdl4

The elimination procedure for the competition number is not optimal

Stephen G. Hartke
2006 Discrete Applied Mathematics  
Given an acyclic digraph D, the competition graph C(D) is defined to be the undirected graph with V (D) as its vertex set and where vertices x and y are adjacent if there exists another vertex z such that  ...  the competition number for all graphs.  ...  Acknowledgments The author would like to thank Fred Roberts and Suh-Ryung Kim for helpful discussions and encouragement and an anonymous referee for suggestions that improved the readability of the paper  ... 
doi:10.1016/j.dam.2005.11.009 fatcat:z4dpj7eac5gcreghdq6iolhrea

Inversions, cuts, and orientations

Wei Ping Liu, Ivan Rival
1991 Discrete Mathematics  
Let P be an n-element ordered set with connected covering graph. Then the number of distinct inversions is at least (n' + 2n)/2 -n log, n.  ...  Let e(n) be the smallest number such that any inversion of P can be produced by pushdowns reversing successively at most e(n) edges. ordered set?  ... 
doi:10.1016/0012-365x(91)90045-4 fatcat:lya4w7aykbb7xlk63wh3r7fv6y

Series-parallel planar ordered sets have pagenumber two [chapter]

Mohammad Alzohairi, Ivan Rival
1997 Lecture Notes in Computer Science  
We say a covers b (or b covered by a) in the ordered set P, and write a ~ b (or b -< a), if whenever a > c > b then c = b.  ...  In a book embedding for an ordered set P the vertices of P on the spine form a linear extension (a total order L = {xl < x2 < ' . . < x~} of the elements of P is a linear extension if x < y in L whenever  ...  Also, they derive a general lower bound on the page number of ordered sets and upper bounds for special classes of ordered sets.  ... 
doi:10.1007/3-540-62495-3_34 fatcat:f6ml5fpxireszje4wl7nivtbom

Enumerating orientations of ordered sets

Wei-Ping Liu, Ivan Rival
1991 Discrete Mathematics  
We exploit independent subsets and matchings to establish further such estimates for certain of the outstanding classes of ordered sets.  ...  Rival, Enumerating orientations of ordered sets, Discrete Mathematics 88 (1991) 239-247. Almost every connected, n-element ordered set has 2"" orientations.  ...  Moreover, we thank one referee for pointing out to use the formulation, especially of Theorem 6, more general than our original one.  ... 
doi:10.1016/0012-365x(91)90012-q fatcat:s3h2drqi5rcrzfic67fcz6crlm

Generalized abc-Block Edge Transformation Graph Q

K.G.Mirajkar, B. Pooja, Shreekant Patil
2019 Zenodo  
A cut vertex of a connected graph is the one whose removal increases the number of components.  ...  Throughout the paper, we only consider simple graphs without isolated vertices. Definitions not given here may be found in [5].  ...  The vertex covering number α 0 (G) of G is the minimum number of vertices in a vertex covering set of G. A set of edges in a graph G is an edge covering set, which covers all vertices of G.  ... 
doi:10.5281/zenodo.3203811 fatcat:the2jbklvrhsvhkseyn6l3wmja

Signed (b,k)-Edge Covers in Graphs

A. N. Ghameshlou, A. Khodkar, R. Saei, S.M. Sheikholeslami
2010 Intelligent Information Management  
Let be a simple graph with vertex set and edge set . Let have at least vertices of degree at least , where and b are positive integers. A function is said to be a signed -edge cover of G if ( , b k)  ...  For an excellent survey of results on edge covers and b-edge covers, see Schrijver [5] . We consider a variant of the standard edge cover problem. Let G be a graph with vertex set and edge set .  ...  A b-edge cover of a graph G is a set C of edges of G such that each vertex of G is incident to at least b edges of C.  ... 
doi:10.4236/iim.2010.22017 fatcat:nbfbrfxwmrbbtela3api354uca

Nonsplit Geodetic Number of a Lict Graph

Venkanagouda MGoudar, Tejaswini K. M., Venkatesha Venkatesha
2014 International Journal of Computer Applications  
The bounds on non split geodetic number in terms of elements of G and covering number of G. Further the relationship between nonsplit geodetic number and geodetic number of a graph is established.  ...  is a geodetic set and S V  is connected. The nonsplit geodetic number of a lict graph   G  , denoted by     G g ns  , is the minimum cardinality of a nonsplit geodetic set of   G  .  ...  The edge covering number   G 1  of a graph G is the minimum cardinality of an edge cover of G.  ... 
doi:10.5120/15360-3836 fatcat:qbwrpiv6sjf2dl2xm6uejduyty

The geodetic vertex covering number of a graph

V.M. Arul Flower Mary, J. Anne Mary Leema, P. Titus, B. Uma Devi
2020 Malaya Journal of Matematik  
The minimum cardinality of a geodetic vertex cover is the geodetic vertex covering number of G denoted by g α (G). Any geodetic vertex cover of cardinality g α (G) is a g α -set of G.  ...  A subset S of vertices in a connected graph G of order at least two is called a geodetic vertex cover if S is both a geodetic set and a vertex covering set.  ...  The vertex covering number of a graph was studied in [7] . A set of vertices (edges) in a graph G is independent if no two of the vertices (edges) are adjacent.  ... 
doi:10.26637/mjm0802/0061 fatcat:2vsbe6ibxncuffklex2afgdbgm

The Covert Set-Cover Problem with Application to Network Discovery [chapter]

Sandeep Sen, V. N. Muralidhara
2010 Lecture Notes in Computer Science  
We want to find a small set-cover using a minimal number of such queries.  ...  We apply this technique to the network discovery problem that involves certifying all the edges and non-edges of an unknown n-vertices graph based on layered-graph queries from a minimal number of vertices  ...  Acknowledgement The first author is thankful to Rajeev Raman and Thomas Erlebach for introducing him to the problem and subsequent technical discussions.  ... 
doi:10.1007/978-3-642-11440-3_21 fatcat:wjkj3sdg4zdppjpcw6rdspusau

On total covering and matching of graphs

A Meir
1978 Journal of combinatorial theory. Series B (Print)  
Let G be a graph with no loops or multiple edges having node set N(G) and edge set E(G). The elements of N(G) u E(G) are called elements of G.  ...  Two elements of G are called independent if neither one covers the other. A set % of elements of G is called a total cover if the elements of %Y cover all elements of G and V is minimal.  ...  Because of the assumed minimality of the number of edges in 4, the node Q must be joined to a node S and S E 9.  ... 
doi:10.1016/0095-8956(78)90017-5 fatcat:rgieppoz5ndx5mpdbokefz2hsu

Fractional Biclique Covers and Partitions of Graphs

Valerie L. Watts
2006 Electronic Journal of Combinatorics  
This paper investigates the fractional biclique cover number, $bc^*(G)$, and the fractional biclique partition number, $bp^*(G)$, of a graph $G$.  ...  It is also shown that $bc^*(G)$ is a better lower bound on the Boolean rank of a binary matrix than the maximum number of isolated ones of the matrix.  ...  Gregory for mentioning the definition of bp * (G), observing that edges in the same orbit of Aut G may be assigned the same weight, and for suggesting Theorem 3.2.  ... 
doi:10.37236/1100 fatcat:4edzt2pcq5ekhe42zzupndtafy

A Generalization of Opsut's Lower Bounds for the Competition Number of a Graph

Yoshio Sano
2012 Graphs and Combinatorics  
The competition graph C(D) of a digraph D is a (simple undirected) graph which has the same vertex set as D and has an edge between two distinct vertices x and y if and only if there exists a vertex v  ...  Roberts defined the competition number k(G) of a graph G as the minimum number of such isolated vertices.  ...  Acknowledgment The author was supported by JSPS Research Fellowships for Young Scientists. The author was also supported partly by Global COE program "Fostering Top Leaders in Mathematics".  ... 
doi:10.1007/s00373-012-1188-5 fatcat:lgot2tjj2jfkdkgs22hdkwf6b4
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