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## Filters

##
###
On the construction of graphs of nullity one

1998
*
Discrete Mathematics
*

For k at most 5,

doi:10.1016/s0012-365x(97)00036-8
fatcat:pvul5knyqvbrbhxj7fnrtzszy4
*the*distinct cores and corresponding minimal configurations (61 in number) are identified. This provides a method*of**constructing*singular*graphs*from others*of*smaller order. ... These correspond to a number k*of*linearly dependent rows*of**the*adjacency matrix determining what is termed as a core*of**the*singular*graph*. ... Acknowledgements*The*author wishes to thank*the*referees for their comments that helped to improve*the*content*of**the*paper. Sciriha/Discrete Mathematics 181 (1998) 193-211 ...##
###
Construction of larger singular and nonsingular graphs using a path

2020
*
Malaya Journal of Matematik
*

As singular

doi:10.26637/mjm0804/0049
fatcat:r3v5hqcarfftpcrmcuawicfq2m
*graphs*have different types*of*vertices,*the**graphs**constructed*in this way differ in*nullity*depending*on**the*vertex we are joining during*construction*. ... An attempt was made to*construct*singular*graph**of*maximum*nullity*. Various spectral properties*of**the*resulting*graphs*were studied. ... Larger singular*graph**constructed*by an even Path*Construction*using*the*path P 2t+1 In this section we*construct**the**graph*KH + P 2t+1 using*the*path P 2t+1*of*odd number*of*vertices. ...##
###
Coalesced and embedded nut graphs in singular graphs

2008
*
Ars Mathematica Contemporanea
*

We investigate how

doi:10.26493/1855-3974.20.7cc
fatcat:jf44brr3oncwhjofcpjhmp7smy
*the*concept*of*nut*graphs*emerges as an underlying theme in*the*theory*of*singular*graphs*. ... Nut*graphs*can be coalesced with singular*graphs*at particular vertices or grown into a family*of*core*graphs**of*larger*nullity*by adding a vertex at a time. ... If*the**nullity*s*of*F is more than*one*, then (ii) and (iii) follow from*the**construction**of*a MC in Definition 7. Conversely, if*the*three conditions hold, then a. ...##
###
Change of nullity of a graph under two operations

2021
*
Maǧallaẗ Al-Kuwayt li-l-ʿulūm
*

Two types

doi:10.48129/kjs.v48i4.9111
fatcat:l2qd6rjoebf3vkzggchaytc5su
*of**graphs*are defined, and*the*change*of*their*nullities*is studied, namely,*the**graph*G+ab*constructed*from G by adding a new vertex ab which is joint to all neighbors*of*both vertices a and ...*The**nullity**of*a*graph*G is*the*multiplicity*of*zero as an eigenvalue*of*its adjacency matrix. ...*The**nullity*, η(G*of*a*graph*G) is*the*multiplicity*of*zero as an eigenvalue*of*its adjacency matrix, see (Cheng & Liu, 2007) ., So a*graph*is singular if its*nullity*is at least*one*. ...##
###
On the nullity of a family of tripartite graphs

2016
*
Acta Universitatis Sapientiae: Informatica
*

*The*eigenvalues

*of*

*the*adjacency matrix

*of*a

*graph*form

*the*spectrum

*of*

*the*

*graph*.

*The*multiplicity

*of*

*the*eigenvalue zero in

*the*spectrum

*of*a

*graph*is called

*nullity*

*of*

*the*

*graph*. ... As observed

*the*

*nullity*problem in tripartite

*graphs*does not follow as an extension to that

*of*

*the*

*nullity*

*of*bipartite

*graphs*, this makes

*the*study

*of*

*nullity*in tripartite

*graphs*interesting. ... [5] and modified

*the*method

*of*

*constructing*

*the*respective trees. Tan and Liu [11] found

*the*

*nullity*set

*of*n-vertex unicyclic

*graphs*with n ≥ 5. ...

##
###
A characterization of singular graphs

2007
*
The Electronic Journal of Linear Algebra
*

A

doi:10.13001/1081-3810.1215
fatcat:iup2lc3iiff75loamjlymnjwca
*graph*G is singular*of**nullity*η(G) ≥ 1, if*the*dimension*of**the*nullspace ker(A)*of*its adjacency matrix A is η(G). ... Characterization*of*singular*graphs*can be reduced to*the*non-trivial solutions*of*a system*of*linear homogeneous equations Ax = 0 for*the*0-1 adjacency matrix A. ...*The*author wishes to thank*the*Universities*of*Malta and Messina,*the*Italian Institute*of*Culture in Malta and*the*American Institute*of*Mathematics, USA, for their support. ...##
###
Existence of regular nut graphs and the fowler construction

2020
*
Applicable Analysis and Discrete Mathematics
*

In this paper

doi:10.2298/aadm190517028g
fatcat:wdegsepkrnb6ddnx2aldqskd2u
*the*problem*of**the*existence*of*regular nut*graphs*is addressed. A generalization*of*Fowler? ... s*Construction*which is a local enlargement applied to a vertex in a*graph*is introduced to generate nut*graphs**of*higher order. Let N (?) ... in*the*number*of*cubic nut*graphs*, and an anonymous referee for suggesting an alternative shorter proof*of*Proposition 11 which we have adopted. ...##
###
On the nullspace of arc-transitive graphs over finite fields

2011
*
Journal of Algebraic Combinatorics
*

We also show that

doi:10.1007/s10801-011-0340-2
fatcat:i33qk4bwujgqhf3tboniol4jwu
*the*p-*nullity**of*a vertextransitive*graph**of*order a power*of*p is zero, provided that*the*valency*of**the**graph*is coprime to p. ...*The**nullity**of*A (that is,*the*dimension*of**the*nullspace*of*A), when viewed as a matrix over a field*of*prime characteristic p, is called*the*p-*nullity**of*Γ . ...*The*second author is supported by UWA as part*of**the*Australian Council Federation Fellowship Project FF0776186. ...##
###
Computing the dependence of graph energy on nullity: The method of siblings

2021
*
Discrete Mathematics Letters
*

*The*energy

*of*a

*graph*is

*the*sum

*of*absolute values

*of*its eigenvalues.

*The*

*nullity*

*of*a

*graph*is

*the*algebraic multiplicity

*of*number zero in its spectrum. ...

*The*dependence

*of*

*graph*energy

*on*

*nullity*is

*of*importance in chemical applications. In this paper we present a method by means

*of*which this dependence can be calculated. ... Thus, in

*the*case

*of*siblings,

*the*energy effect

*of*

*nullity*can be straightforwardly evaluated by calculating two ordinary

*graph*energies.

*Graph*siblings are relatively easy to

*construct*. ...

##
###
Trees with maximum nullity

2005
*
Linear Algebra and its Applications
*

*The*

*nullity*

*of*a

*graph*is

*the*multiplicity

*of*

*the*eigenvalue zero in its spectrum. Among nvertex trees,

*the*star has greatest

*nullity*(equal to n − 2). ... Methods for

*constructing*such maximum-

*nullity*trees are described. ...

*The*number

*of*zero eigenvalues in

*the*spectrum

*of*

*the*

*graph*G is called its

*nullity*and is denoted by η(G). ...

##
###
Maximum and minimum nullity of a tree degree sequence
[article]

2018
*
arXiv
*
pre-print

*The*

*nullity*

*of*a

*graph*is

*the*multiplicity

*of*

*the*eigenvalue zero in its adjacency spectrum. ... Algorithms for

*constructing*such minimum-

*nullity*and maximum-

*nullity*trees are described. ... Acknowledgement

*The*authors would like to thank to Vilmar Trevisan, Emilio Allem, Rodrigo Orsini Braga and Maikon Machado Toledo

*of*Universidade Federal do Rio Grande do Sul for a wonderful time in our ...

##
###
Most Clicks Problem in Lights Out
[article]

2022
*
arXiv
*
pre-print

*The*answer to

*the*MCP is already known for

*nullity*0

*graphs*: those

*on*which every initial configuration is solvable. ... Consider a game played

*on*a simple

*graph*G = (V, E) where each vertex consists

*of*a clickable light. Clicking any vertex v toggles

*the*

*on*/off state

*of*v and its neighbors. ... A Lower Bound

*on*

*Nullity*For each even parity cover

*of*a smaller grid, we can

*construct*a unique even parity cover

*of*a larger grid. ...

##
###
On the nullity of graphs with pendent vertices

2008
*
Linear Algebra and its Applications
*

*The*

*nullity*

*of*a

*graph*G, denoted by η(G), is

*the*multiplicity

*of*

*the*eigenvalue zero in its spectrum. Cheng and Liu [B. Cheng, B. Liu,

*On*

*the*

*nullity*

*of*

*graphs*, Electron. J. ... As a consequence we, respectively, determine

*the*

*nullity*sets

*of*trees, unicyclic

*graphs*, bicyclic

*graphs*and tricyclic

*graphs*

*on*n vertices. ... Acknowledgments

*The*author is grateful to

*the*referees for their valuable comments, corrections and suggestions, especially for

*one*

*of*

*the*referees who called

*the*author's attention to Refs. ...

##
###
Omni-conducting and omni-insulating molecules

2014
*
Journal of Chemical Physics
*

*The*class

*of*conduction behaviour depends critically

*on*

*the*number

*of*non-bonding orbitals (NBO)

*of*

*the*molecular system (corresponding to

*the*

*nullity*

*of*

*the*

*graph*). ... Distinct omni-conductors with a single NBO are all also strong and correspond exactly to

*the*class

*of*

*graphs*known as nut

*graphs*. ...

*Graphs*

*of*

*nullity*g s = 1 For

*graphs*

*of*

*nullity*

*one*, there is an analogous but weaker condition for conductivity, based

*on*

*the*adjugate matrix. Theorem 4.2. ...

##
###
Non-Separable and Planar Graphs

1932
*
Transactions of the American Mathematical Society
*

By Theorem 13, this is

doi:10.2307/1989545
fatcat:2jesblxchrg65kgkhvar6icgeq
*the*sum*of**the**nullities**of*77i, • • ■ , 77m_i. Therefore*the**nullity**of*some*one**of*these*graphs*, say 77,-, is >0. Suppose first*the**nullity**of*Hi is 1. ... For let us*construct*G arc by arc Each time we add an arc joining two separate pieces,*the**nullity*and*the*number*of*regiong remain*the*same. ...
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