scholarly journals On Degree Properties of Crossing-Critical Families of Graphs

10.37236/7753 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Drago Bokal ◽  
Mojca Bračič ◽  
Marek Derňár ◽  
Petr Hliněný
Keyword(s):  
Average Degree ◽  
Critical Graphs ◽  
Odd Degree ◽  
Vertex Degrees ◽  
Open Question

Answering an open question from 2007, we construct infinite $k$-crossing-critical families of graphs that contain vertices of any prescribed odd degree, for any sufficiently large $k$. To answer this question, we introduce several properties of infinite families of graphs and operations on the families allowing us to obtain new families preserving those properties. This conceptual setup allows us to answer general questions on behaviour of degrees in crossing-critical graphs: we show that, for any set of integers $D$ such that $\min(D)\geq 3$ and $3,4\in D$, and for any sufficiently large $k$, there exists a $k$-crossing-critical family such that the numbers in $D$ are precisely the vertex degrees that occur arbitrarily often in (large enough) graphs of this family. Furthermore, even if both $D$ and some average degree in the interval $(3,6)$ are prescribed, $k$-crossing-critical families exist for any sufficiently large $k$.

scholarly journals On the average degree of critical graphs with maximum degree six

2011 ◽  
Vol 311 (21) ◽  
pp. 2574-2576 ◽  
Author(s):  
Lianying Miao ◽  
Jibin Qu ◽  
Qingbo Sun
Keyword(s):  
Maximum Degree ◽  
Average Degree ◽  
Critical Graphs

scholarly journals A Better Lower Bound on Average Degree of Online $k$-List-Critical Graphs

10.37236/6405 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Landon Rabern
Keyword(s):  
Lower Bound ◽  
Average Degree ◽  
Critical Graphs ◽  
Critical Graph

We improve the best known bounds on average degree of online $k$-list-critical graphs for $k \geqslant 6$. Specifically, for $k \geqslant 7$ we show that every non-complete online $k$-list-critical graph has average degree at least $k-1 + \frac{(k-3)^2 (2 k-3)}{k^4-2 k^3-11 k^2+28 k-14}$ and every non-complete online $6$-list-critical graph has average degree at least $5 + \frac{93}{766}$. The same bounds hold for offline $k$-list-critical graphs.

scholarly journals A Better Lower Bound on Average Degree of 4-List-Critical Graphs

10.37236/5971 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Landon Rabern
Keyword(s):  
Lower Bound ◽  
Short Note ◽  
Average Degree ◽  
Critical Graphs ◽  
Critical Graph

This short note proves that every non-complete $k$-list-critical graph has average degree at least $k-1 + \frac{k-3}{k^2-2k+2}$. This improves the best known bound for $k = 4,5,6$. The same bound holds for online $k$-list-critical graphs.

scholarly journals New Infinite Families of Almost-Planar Crossing-Critical Graphs

10.37236/826 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Petr Hliněný
Keyword(s):  
Average Degree ◽  
The Other ◽  
Critical Graphs

We show that, for all choices of integers $k>2$ and $m$, there are simple $3$-connected $k$-crossing-critical graphs containing more than $m$ vertices of each even degree $\leq2k-2$. This construction answers one half of a question raised by Bokal, while the other half asking analogously about vertices of odd degrees at least $7$ in crossing-critical graphs remains open. Furthermore, our newly constructed graphs have several other interesting properties; for instance, they are almost planar and their average degree can attain any rational value in the interval $\big[3+{1\over5},6-{8\over k+1}\big)$.

scholarly journals Book Embedding of Infinite Family ((2h+3 2))-Crossing-Critical Graphs for h=1 with Rational Average Degree r∈(3.5,4)

d'CARTESIAN ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 145
Author(s):  
Sheren H. Wilar ◽  
Benny Pinontoan ◽  
Chriestie E.J.C. Montolalu
Keyword(s):  
Intersection Number ◽  
Infinite Family ◽  
Infinite Graph ◽  
Average Degree ◽  
Critical Graphs ◽  
Book Embedding ◽  
Embed Graph ◽  
Principal Tool ◽  
Page Book

A principal tool used in construction of crossing-critical graphs are tiles. In the tile concept, tiles can be arranged by gluing one tile to another in a linear or circular fashion. The series of tiles with circular fashion form an infinite graph family. In this way, the intersection number of this family of graphs can be determined. In this research, has been formed an infinite family graphs Q_((1,s,b) ) (n) with average degree r between 3.5 and 4. The graph formed by gluing together many copies of the tile P_((1,s,b) ) in circular fashion, where the tile P_((1,s,b) ) consist of two identical pieces of tile. And then, the graph embedded into the book to determine the pagenumber that can be formed. When embed graph into book, the vertices are put on a line called the spine and the edges are put on half-planes called the pages. The results obtained show that the graph Q_((1,s,b) ) (n) has 10-crossing-critical and book embedding of graph has 4-page book.

scholarly journals Multipartite Separability of Laplacian Matrices of Graphs

10.37236/150 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Chai Wah Wu
Keyword(s):  
Quantum Mechanics ◽  
Physical Properties ◽  
Laplacian Matrix ◽  
Bipartite Graphs ◽  
Density Matrices ◽  
Laplacian Matrices ◽  
Complete Bipartite Graphs ◽  
Vertex Degrees ◽  
Open Question ◽  
Product Of Graphs

Recently, Braunstein et al. introduced normalized Laplacian matrices of graphs as density matrices in quantum mechanics and studied the relationships between quantum physical properties and graph theoretical properties of the underlying graphs. We provide further results on the multipartite separability of Laplacian matrices of graphs. In particular, we identify complete bipartite graphs whose normalized Laplacian matrix is multipartite entangled under any vertex labeling. Furthermore, we give conditions on the vertex degrees such that there is a vertex labeling under which the normalized Laplacian matrix is entangled. These results address an open question raised in Braunstein et al. Finally, we show that the Laplacian matrix of any product of graphs (strong, Cartesian, tensor, lexicographical, etc.) is multipartite separable, extending analogous results for bipartite and tripartite separability.

scholarly journals On Omega Index and Average Degree of Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Sadik Delen ◽  
Musa Demirci ◽  
Ahmet Sinan Cevik ◽  
Ismail Naci Cangul
Keyword(s):  
Social Sciences ◽  
Arithmetic Mean ◽  
Average Degree ◽  
Graph Invariant ◽  
Edge Deletion ◽  
Graph Operations ◽  
Vertex Deletion ◽  
Tricyclic Graphs ◽  
Cyclic Graphs ◽  
Vertex Degrees

Average degree of a graph is defined to be a graph invariant equal to the arithmetic mean of all vertex degrees and has many applications, especially in determining the irregularity degrees of networks and social sciences. In this study, some properties of average degree have been studied. Effect of vertex deletion on this degree has been determined and a new proof of the handshaking lemma has been given. Using a recently defined graph index called o m e g a index, average degree of trees, unicyclic, bicyclic, and tricyclic graphs have been given, and these have been generalized to k -cyclic graphs. Also, the effect of edge deletion has been calculated. The average degree of some derived graphs and some graph operations have been determined.

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