Generalization of the Symmetric Starter of the Orthogonal Double Covers of the Complete Bipartite Graph

An orthogonal double cover (ODC) of the complete graph is a collection of graphs such that every two of them share exactly one edge and every edge of the complete graph belongs to exactly two of the graphs. In this paper, we consider the case where the graph to be covered twice is the complete bipartite graphKmn,mn(for any values ofm,n) and all graphs in the collection are isomorphic to certain spanning subgraphs. Furthermore, the ODCs ofKn,nby certain disjoint stars are constructed.

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Graceful Labeling for Some Complete Bipartite Graph

Journal of Computer and Mathematical Sciences ◽

10.29055/jcms/960 ◽

2018 ◽

Vol 9 (12) ◽

pp. 2147-2152

Author(s):

V. Raju ◽

M. Paruvatha vathana

Keyword(s):

Bipartite Graph ◽

Complete Bipartite Graph ◽

Graceful Labeling

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On the Crossing Number of $K_{m,n}$

The Electronic Journal of Combinatorics ◽

10.37236/1748 ◽

2003 ◽

Vol 10 (1) ◽

Cited By ~ 4

Author(s):

Nagi H. Nahas

Keyword(s):

Lower Bound ◽

Bipartite Graph ◽

Complete Bipartite Graph ◽

Crossing Number

The best lower bound known on the crossing number of the complete bipartite graph is : $$cr(K_{m,n}) \geq (1/5)(m)(m-1)\lfloor n/2 \rfloor \lfloor(n-1)/2\rfloor$$ In this paper we prove that: $$cr(K_{m,n}) \geq (1/5)m(m-1)\lfloor n/2 \rfloor \lfloor (n-1)/2 \rfloor + 9.9 \times 10^{-6} m^2n^2$$ for sufficiently large $m$ and $n$.

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Join Products K2,3 + Cn

Mathematics ◽

10.3390/math8060925 ◽

2020 ◽

Vol 8 (6) ◽

pp. 925

Author(s):

Michal Staš

Keyword(s):

Bipartite Graph ◽

Complete Bipartite Graph ◽

Crossing Number ◽

Arithmetic Means ◽

Minimum Number ◽

Edge Crossings

The crossing number cr ( G ) of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main goal of the paper is to state the crossing number of the join product K 2 , 3 + C n for the complete bipartite graph K 2 , 3 , where C n is the cycle on n vertices. In the proofs, the idea of a minimum number of crossings between two distinct configurations in the various forms of arithmetic means will be extended. Finally, adding one more edge to the graph K 2 , 3 , we also offer the crossing number of the join product of one other graph with the cycle C n .

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Spectral Gap of the Largest Eigenvalue of the Normalized Graph Laplacian

Communications in Mathematics and Statistics ◽

10.1007/s40304-020-00222-7 ◽

2021 ◽

Author(s):

Jürgen Jost ◽

Raffaella Mulas ◽

Florentin Münch

Keyword(s):

Lower Bound ◽

Bipartite Graph ◽

Spectral Gap ◽

Complete Bipartite Graph ◽

Graph Laplacian ◽

New Method ◽

Single Edge ◽

Largest Eigenvalue ◽

Complement Graph ◽

The Largest Eigenvalue

AbstractWe offer a new method for proving that the maxima eigenvalue of the normalized graph Laplacian of a graph with n vertices is at least $$\frac{n+1}{n-1}$$ n + 1 n - 1 provided the graph is not complete and that equality is attained if and only if the complement graph is a single edge or a complete bipartite graph with both parts of size $$\frac{n-1}{2}$$ n - 1 2 . With the same method, we also prove a new lower bound to the largest eigenvalue in terms of the minimum vertex degree, provided this is at most $$\frac{n-1}{2}$$ n - 1 2 .

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