scholarly journals End-Completely-Regular and End-Inverse Lexicographic Products of Graphs

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Hailong Hou ◽  
Rui Gu
Keyword(s):  
Bipartite Graphs ◽  
Endomorphism Monoid ◽  
Regular Graphs ◽  
Completely Regular

A graphXis said to be End-completely-regular (resp., End-inverse) if its endomorphism monoid End(X) is completely regular (resp., inverse). In this paper, we will show that ifX[Y] is End-completely-regular (resp., End-inverse), then bothXandYare End-completely-regular (resp., End-inverse). We give several approaches to construct new End-completely-regular graphs by means of the lexicographic products of two graphs with certain conditions. In particular, we determine the End-completely-regular and End-inverse lexicographic products of bipartite graphs.

scholarly journals Cycle partitions of regular graphs

2020 ◽  
pp. 1-24
Author(s):  
Vytautas Gruslys ◽  
Shoham Letzter
Keyword(s):  
Regular Graph ◽  
Bipartite Graphs ◽  
Disjoint Paths ◽  
Regular Graphs ◽  
Simple Construction ◽  
Vertex Set ◽  
Almost All ◽  
Vertex Disjoint

Abstract Magnant and Martin conjectured that the vertex set of any d-regular graph G on n vertices can be partitioned into $n / (d+1)$ paths (there exists a simple construction showing that this bound would be best possible). We prove this conjecture when $d = \Omega(n)$ , improving a result of Han, who showed that in this range almost all vertices of G can be covered by $n / (d+1) + 1$ vertex-disjoint paths. In fact our proof gives a partition of V(G) into cycles. We also show that, if $d = \Omega(n)$ and G is bipartite, then V(G) can be partitioned into n/(2d) paths (this bound is tight for bipartite graphs).

scholarly journals Can Colour-Blind Distinguish Colour Palettes?

10.37236/2319 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Rafał Kalinowski ◽  
Monika Pilśniak ◽  
Jakub Przybyło ◽  
Mariusz Woźniak
Keyword(s):  
Large Degree ◽  
Bipartite Graphs ◽  
Complete Graphs ◽  
Regular Graphs ◽  
Lovász Local Lemma ◽  
Local Lemma ◽  
Comprehensive Information ◽  
Lovasz Local Lemma ◽  
Delta G ◽  
Irregular Graphs

Let $c:E(G)\rightarrow [k]$ be  a colouring, not necessarily proper, of edges of a graph $G$. For a vertex $v\in V$, let $\overline{c}(v)=(a_1,\ldots,a_k)$, where $ a_i =|\{u:uv\in E(G),\;c(uv)=i\}|$, for $i\in [k].$ If we re-order the sequence $\overline{c}(v)$ non-decreasingly, we obtain a sequence $c^*(v)=(d_1,\ldots,d_k)$, called a palette of a vertex $v$. This can be viewed as the most comprehensive information about colours incident with $v$ which can be delivered by a person who is unable to name colours but distinguishes one from another. The smallest $k$ such that $c^*$ is a proper colouring of vertices of $G$ is called the colour-blind index of a graph $G$, and is denoted by dal$(G)$. We conjecture that there is a constant $K$ such that dal$(G)\leq K$ for every graph $G$ for which the parameter is well defined. As our main result we prove that $K\leq 6$ for regular graphs of sufficiently large degree, and for irregular graphs with $\delta (G)$ and $\Delta(G)$ satisfying certain conditions. The proofs are based on the Lopsided Lovász Local Lemma. We also show that $K=3$ for all regular bipartite graphs, and for complete graphs of order $n\geq 8$.

scholarly journals The Entropy of Weighted Graphs with Atomic Bond Connectivity Edge Weights

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Young Chel Kwun ◽  
Hafiz Mutee ur Rehman ◽  
Muhammad Yousaf ◽  
Waqas Nazeer ◽  
Shin Min Kang
Keyword(s):  
Open Problem ◽  
Bipartite Graphs ◽  
Regular Graphs ◽  
Weighted Graphs ◽  
Connected Graphs ◽  
Unicyclic Graphs ◽  
Star Graphs ◽  
Chemical Graphs ◽  
Complete Bipartite Graphs

The aim of this report to solve the open problem suggested by Chen et al. We study the graph entropy with ABC edge weights and present bounds of it for connected graphs, regular graphs, complete bipartite graphs, chemical graphs, tree, unicyclic graphs, and star graphs. Moreover, we compute the graph entropy for some families of dendrimers.

scholarly journals End-regular and End-orthodox generalized lexicographic products of bipartite graphs

2016 ◽  
Vol 14 (1) ◽  
pp. 229-236 ◽  
Author(s):  
Rui Gu ◽  
Hailong Hou
Keyword(s):  
Orthodox Semigroup ◽  
Bipartite Graphs ◽  
Endomorphism Monoid

AbstractA graph X is said to be End-regular (End-orthodox) if its endomorphism monoid End(X) is a regular (orthodox) semigroup. In this paper, we determine the End-regular and the End-orthodox generalized lexicographic products of bipartite graphs.

scholarly journals Families of nested completely regular codes and distance-regular graphs

2015 ◽  
Vol 9 (2) ◽  
pp. 233-246 ◽  
Author(s):  
Joaquim Borges ◽  
◽  
Josep Rifà ◽  
Victor A. Zinoviev ◽  
◽  
...  
Keyword(s):  
Regular Graphs ◽  
Completely Regular ◽  
Distance Regular Graphs ◽  
Regular Codes ◽  
Completely Regular Codes

scholarly journals Some Families of Orthogonal Polynomials of a Discrete Variable and their Applications to Graphs and Codes

10.37236/172 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
M. Cámara ◽  
J. Fàbrega ◽  
M. A. Fiol ◽  
E. Garriga
Keyword(s):  
Orthogonal Polynomials ◽  
Regular Graphs ◽  
Discrete Variable ◽  
Completely Regular ◽  
Local Distance ◽  
Distance Regular Graphs ◽  
Regular Codes ◽  
Completely Regular Codes ◽  
Orthogonal Systems ◽  
Spectral Characterizations

We present some related families of orthogonal polynomials of a discrete variable and survey some of their applications in the study of (distance-regular) graphs and (completely regular) codes. One of the main peculiarities of such orthogonal systems is their non-standard normalization condition, requiring that the square norm of each polynomial must equal its value at a given point of the mesh. For instance, when they are defined from the spectrum of a graph, one of these families is the system of the predistance polynomials which, in the case of distance-regular graphs, turns out to be the sequence of distance polynomials. The applications range from (quasi-spectral) characterizations of distance-regular graphs, walk-regular graphs, local distance-regularity and completely regular codes, to some results on representation theory.

scholarly journals Differentials in certain classes of graphs

2010 ◽  
Vol 41 (2) ◽  
pp. 129-138 ◽  
Author(s):  
P. Roushini Leely Pushpam ◽  
D. Yokesh
Keyword(s):  
Binary Tree ◽  
Maximum Degree ◽  
Bipartite Graphs ◽  
Regular Graphs ◽  
Complete Binary Tree ◽  
Unicyclic Graphs ◽  
Grid Graphs ◽  
Split Graphs ◽  
Delta G

Let $X subset V$ be a set of vertices in a graph $G = (V, E)$. The boundary $B(X)$ of $X$ is defined to be the set of vertices in $V-X$ dominated by vertices in $X$, that is, $B(X) = (V-X) cap N(X)$. The differential $ partial(X)$ of $X$ equals the value $ partial(X) = |B(X)| - |X|$. The differential of a graph $G$ is defined as $ partial(G) = max { partial(X) | X subset V }$. It is easy to see that for any graph $G$ having vertices of maximum degree $ Delta(G)$, $ partial(G) geq Delta (G) -1$. In this paper we characterize the classes of unicyclic graphs, split graphs, grid graphs, $k$-regular graphs, for $k leq 4$, and bipartite graphs for which $ partial(G) = Delta(G)-1$. We also determine the value of $ partial(T)$ for any complete binary tree $T$.

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