scholarly journals A Construction of Cospectral Graphs for the Normalized Laplacian

10.37236/718 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Steve Butler ◽  
Jason Grout
Keyword(s):  
Bipartite Graph ◽  
Related Result ◽  
Bipartite Graphs ◽  
Special Structure ◽  
Regular Graphs ◽  
Cospectral Graphs ◽  
The Matrix ◽  
Local Modification ◽  
Large Families

We give a method to construct cospectral graphs for the normalized Laplacian by a local modification in some graphs with special structure. Namely, under some simple assumptions, we can replace a small bipartite graph with a cospectral mate without changing the spectrum of the entire graph. We also consider a related result for swapping out biregular bipartite graphs for the matrix $A+tD$. We produce (exponentially) large families of non-bipartite, non-regular graphs which are mutually cospectral, and also give an example of a graph which is cospectral with its complement but is not self-complementary.

scholarly journals On Kites, Comets, and Stars. Sums of Eigenvector Coefficients in (Molecular) Graphs

2002 ◽  
Vol 57 (3-4) ◽  
pp. 143-153 ◽  
Author(s):  
Gerta Rücker ◽  
Christoph Rücker ◽  
Ivan Gutman
Keyword(s):  
Bipartite Graph ◽  
Bipartite Graphs ◽  
Computer Search ◽  
Regular Graphs ◽  
Graph Invariants ◽  
Star Graphs ◽  
Absolute Value ◽  
Tree Graphs ◽  
Discriminating Power ◽  
Approximate Equations

Two graph invariants were encountered that form the link between (molecular) walk counts and eigenvalues of graph adjacency matrices. In particular, the absolute value of the sum of coefficients of the first or principal (normalized) eigenvector, s1, and the analogous quantity sn, pertaining to the last eigenvector, appear in equations describing some limits (for infinitely long walks) of relative frequencies of several walk counts. Quantity s1 is interpreted as a measure of mixedness of a graph, and sn, which plays a role for bipartite graphs only, is interpreted as a measure of the imbalance of a bipartite graph. Consequently, sn is maximal for star graphs, while the minimal value of sn is zero. Mixedness s1 is maximal for regular graphs. Minimal values of s1 were found by exhaustive computer search within the sample of all simple connected undirected n-vertex graphs, n≤10: They are encountered among graphs called kites. Within the special sample of tree graphs (searched for n≤20) so-called double snakes have maximal s1, while the trees with minimal s1 are so-called comets. The behaviour of stars and double snakes can be described by exact equations, while approximate equations for s1 of kites and comets could be derived that are fully compatible with and allow to predict some pecularities of the results of the computer search. Finally, the discriminating power of s1, determined within trees and 4-trees (alkanes), was found to be high.

scholarly journals Some Properties of Regular Line Graphs

2008 ◽  
pp. 44-49
Keyword(s):  
Key Words ◽  
Bipartite Graph ◽  
Line Graph ◽  
Bipartite Graphs ◽  
Line Graphs ◽  
Sufficient Condition ◽  
Regular Graphs ◽  
Necessary And Sufficient Condition ◽  
Regular Line ◽  
Necessary And Sufficient

In this paper, the concept of regular line graph has been introduced. The maximum number of vertices with different degrees in the regular line graphs has also been studied. Further, the necessary and sufficient condition for regular line graph to be bipartite graph have also been proved. Key words: Line Graphs, Regular graphs, Connected graphs, 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 On the induced matching problem in hamiltonian bipartite graphs

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Yinglei Song
Keyword(s):  
Bipartite Graph ◽  
Parameterized Complexity ◽  
Bipartite Graphs ◽  
Matching Problem ◽  
Induced Matching ◽  
Subexponential Time ◽  
Maximum Induced Matching

Abstract In this paper, we study the parameterized complexity of the induced matching problem in hamiltonian bipartite graphs and the inapproximability of the maximum induced matching problem in hamiltonian bipartite graphs. We show that, given a hamiltonian bipartite graph, the induced matching problem is W[1]-hard and cannot be solved in time n o ⁢ ( k ) {n^{o(\sqrt{k})}} , where n is the number of vertices in the graph, unless the 3SAT problem can be solved in subexponential time. In addition, we show that unless NP = P {\operatorname{NP}=\operatorname{P}} , a maximum induced matching in a hamiltonian bipartite graph cannot be approximated within a ratio of n 1 / 4 - ϵ {n^{1/4-\epsilon}} , where n is the number of vertices in the graph.

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 $n\geq 8$.

Interfaces in Fibre Reinforced Cements

1987 ◽  
Vol 114 ◽  
Author(s):  
A. Bentur
Keyword(s):  
Mechanical Performance ◽  
Fibre Surface ◽  
Special Structure ◽  
Analytical Models ◽  
The Matrix

ABSTRACTThe microstructure of the matrix in the vicinity of the fibre surface is quite different from the microstructure of the bulk paste matrix. This can have an important effect on the processes that take place at the interface, such as crack-fibre interaction and debonding. The present paper describes the special structure of this zone, in monofilament and bundled fibre reinforced cements, and discusses its effects on some characteristics of the mechanical performance of the composites, which cannot be predicted by analytical models assuming a uniform matrix up to the fibre surface. The modification of the microstructure at the interface as a means for improving properties in some composites is described.

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.

Rainbow Perfect Matchings in Complete Bipartite Graphs: Existence and Counting

2013 ◽  
Vol 22 (5) ◽  
pp. 783-799 ◽  
Author(s):  
GUILLEM PERARNAU ◽  
ORIOL SERRA
Keyword(s):  
Bipartite Graph ◽  
High Probability ◽  
Perfect Matching ◽  
Complete Bipartite Graph ◽  
Bipartite Graphs ◽  
Perfect Matchings ◽  
Asymptotic Enumeration ◽  
Complete Bipartite Graphs ◽  
Edge Colourings ◽  
Square Matrix

A perfect matchingMin an edge-coloured complete bipartite graphKn,nis rainbow if no pair of edges inMhave the same colour. We obtain asymptotic enumeration results for the number of rainbow perfect matchings in terms of the maximum number of occurrences of each colour. We also consider two natural models of random edge-colourings ofKn,nand show that if the number of colours is at leastn, then there is with high probability a rainbow perfect matching. This in particular shows that almost every square matrix of ordernin which every entry appearsntimes has a Latin transversal.

Finite locally-primitive complete bipartite graphs

2014 ◽  
Vol 17 (1) ◽  
Author(s):  
Wenwen Fan ◽  
Cai Heng Li ◽  
Jiangmin Pan
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Bipartite Graphs ◽  
Complete Bipartite Graphs

Abstract.We characterize groups which act locally-primitively on a complete bipartite graph. The result particularly determines certain interesting factorizations of groups.

scholarly journals Longest cycles in certain bipartite graphs

1998 ◽  
Vol 21 (1) ◽  
pp. 103-106
Author(s):  
Pak-Ken Wong
Keyword(s):  
Bipartite Graph ◽  
Bipartite Graphs ◽  
Longest Cycles

LetGbe a connected bipartite graph with bipartition(X,Y)such that|X|≥|Y|(≥2),n=|X|andm=|Y|. Suppose, for all verticesx∈Xandy∈Y,dist(x,y)=3impliesd(x)+d(y)≥n+1. ThenGcontains a cycle of length2m. In particular, ifm=n, thenGis hamiltomian.

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