A new upper bound for the independence number of edge chromatic critical graphs

2010 ◽  
Vol 68 (3) ◽  
pp. 202-212 ◽  
Author(s):  
Rong Luo ◽  
Yue Zhao
Keyword(s):  
Upper Bound ◽  
Independence Number ◽  
Critical Graphs

scholarly journals New Bounds for the α-Indices of Graphs

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1668
Author(s):  
Eber Lenes ◽  
Exequiel Mallea-Zepeda ◽  
Jonnathan Rodríguez
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Upper Bound ◽  
Adjacency Matrix ◽  
Independence Number ◽  
Minimal Degree ◽  
Diagonal Matrix ◽  
The Matrix

Let G be a graph, for any real 0≤α≤1, Nikiforov defines the matrix Aα(G) as Aα(G)=αD(G)+(1−α)A(G), where A(G) and D(G) are the adjacency matrix and diagonal matrix of degrees of the vertices of G. This paper presents some extremal results about the spectral radius ρα(G) of the matrix Aα(G). In particular, we give a lower bound on the spectral radius ρα(G) in terms of order and independence number. In addition, we obtain an upper bound for the spectral radius ρα(G) in terms of order and minimal degree. Furthermore, for n>l>0 and 1≤p≤⌊n−l2⌋, let Gp≅Kl∨(Kp∪Kn−p−l) be the graph obtained from the graphs Kl and Kp∪Kn−p−l and edges connecting each vertex of Kl with every vertex of Kp∪Kn−p−l. We prove that ρα(Gp+1)<ρα(Gp) for 1≤p≤⌊n−l2⌋−1.

scholarly journals Cubic Graphs with Small Independence Ratio

10.37236/7272 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
József Balogh ◽  
Alexandr Kostochka ◽  
Xujun Liu
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Independence Number ◽  
Regular Graphs ◽  
Cubic Graphs

Let $i(r,g)$ denote the infimum of the ratio $\frac{\alpha(G)}{|V(G)|}$ over the $r$-regular graphs of girth at least $g$, where $\alpha(G)$ is the independence number of $G$, and  let $i(r,\infty) := \lim\limits_{g \to \infty} i(r,g)$. Recently, several new lower bounds of $i(3,\infty)$ were obtained. In particular, Hoppen and Wormald showed in 2015 that $i(3, \infty) \geqslant 0.4375,$ and Csóka improved it to $i(3,\infty) \geqslant 0.44533$ in 2016. Bollobás proved the upper bound  $i(3,\infty) < \frac{6}{13}$  in 1981, and McKay improved it to $i(3,\infty) < 0.45537$in 1987. There were no improvements since then. In this paper, we improve the upper bound to $i(3,\infty) \leqslant 0.454.$

scholarly journals On the Independence Numbers of the Cubes of Odd Cycles

10.37236/2598 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Tom Bohman ◽  
Ron Holzman ◽  
Venkatesh Natarajan
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Independence Number ◽  
Odd Cycle ◽  
Odd Cycles ◽  
Independence Numbers

We give an upper bound on the independence number of the cube of the odd cycle $C_{8n+5}$. The best known lower bound is conjectured to be the truth; we prove the conjecture in the case $8n+5$ prime and, within $2$, for general $n$.

Independence number of generalized products of graphs

2018 ◽  
Vol 13 (01) ◽  
pp. 2050013
Author(s):  
H. S. Mehta ◽  
U. P. Acharya
Keyword(s):  
Lower Bound ◽  
Tensor Product ◽  
Upper Bound ◽  
Independence Number ◽  
Cartesian Product ◽  
Product Formula ◽  
Graph Products ◽  
Graph Parameters

The tensor product and the Cartesian product of two graphs are very well-known graph products and studied in detail. Many graph parameters, particularly independence number, have been studied for these graph products. These two graph products have been generalized by [Formula: see text]-tensor product and [Formula: see text]-Cartesian product, respectively, and studied in detail. In this paper, we discuss the independence number for [Formula: see text]-tensor product [Formula: see text] and [Formula: see text]-Cartesian product [Formula: see text]. In general, we obtain lower bound and upper bound for the independence number.

Independence number and minimum degree for fractional ID-k-factor-critical graphs

2012 ◽  
Vol 84 (1-2) ◽  
pp. 71-76 ◽  
Author(s):  
Sizhong Zhou ◽  
Lan Xu ◽  
Zhiren Sun
Keyword(s):  
Minimum Degree ◽  
Independence Number ◽  
Critical Graphs ◽  
K Factor

Bounds for the Independence Number of Critical Graphs

2000 ◽  
Vol 32 (2) ◽  
pp. 137-140 ◽  
Author(s):  
Gunnar Brinkmann ◽  
Sheshayya A. Choudum ◽  
Stefan Grünewald ◽  
Eckhard Steffen
Keyword(s):  
Independence Number ◽  
Critical Graphs

scholarly journals On the independence number of edge chromatic critical graphs

2014 ◽  
Vol 34 (3) ◽  
pp. 577 ◽  
Author(s):  
Lianying Miao ◽  
Zhengke Miao ◽  
Shiyou Pang ◽  
Wenyao Song
Keyword(s):  
Independence Number ◽  
Critical Graphs
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