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More Tales of Hoffman: bounds for the vector chromatic number of a graph
[article]

2020
*
arXiv
*
pre-print

Let χ(G) denote

arXiv:1812.02613v3
fatcat:es5rf2iupnetzh4mfdqckna524
*the*chromatic*number**of**a**graph*and χ_v(G) denote*the*vector chromatic*number*. For all*graphs*χ_v(G) <χ(G) and for some*graphs*χ_v(G) ≪χ(G). ... We then use one*of*these bounds to derive*a*new characterization*of*χ_v(G). ...*The**Hoffman*bound for χ v (G), proved by Galtman and Bilu, follows when D is*the*zero matrix. Extremal*graphs**A**graph*, G, is said to have*a**Hoffman*coloring if χ(G) equals*the**Hoffman*bound. ...##
###
Sesqui-regular graphs with fixed smallest eigenvalue
[article]

2021
*
arXiv
*
pre-print

In this paper, we will study

arXiv:1904.01274v2
fatcat:n3ewqfq5gza2nns2b6smhfboti
*a*new class*of*regular*graphs*called sesqui-regular*graphs*, which contains strongly regular*graphs*as*a*subclass, and prove that for*a*sesqui-regular*graph*with parameters ... For strongly regular*graphs*with parameters (v, k,*a*,c) and smallest eigenvalue -λ, Neumaier gave two bounds on c by using algebraic property*of*strongly regular*graphs*. ... We are also grateful to*the*referee for his/her careful reading and valuable comments. ...##
###
On the order of regular graphs with fixed second largest eigenvalue
[article]

2018
*
arXiv
*
pre-print

Let v(k, λ) be

arXiv:1809.01888v1
fatcat:uggmkos7nzggtnl226fjyb2nfi
*the*maximum*number**of*vertices*of**a*connected k-regular*graph*with second largest eigenvalue at most λ.*The*Alon-Boppana Theorem implies that v(k, λ) is finite when k > λ^2 + 4/4. ... In this paper, we show that for fixed λ≥1, there exists*a*constant C(λ) such that 2k+2 ≤ v(k, λ) ≤ 2k + C(λ) when k > λ^2 + 4/4. ... Then there exists*a*real*number*C 3 (λ) such that if G has smallest eigenvalue at least −λ, then c 2 ≤ C 3 (λ) or G is*a*complete multipartite*graph*. , then G has diameter 2 by Proposition 1.3. ...##
###
On graphs with smallest eigenvalue at least -3 and their lattices
[article]

2018
*
arXiv
*
pre-print

This result generalizes

arXiv:1804.00369v1
fatcat:dvvlmudxsbcaneappmk2fbsrse
*a*1977 result*of**Hoffman*for connected*graphs*with smallest eigenvalue at least -2. ... In this paper, we show that*a*connected*graph*with smallest eigenvalue at least -3 and large enough minimal degree is 2-integrable. ...*The*real*number*ε can be found as*the*largest*number*among*the*smallest eigenvalues*of**the*minimal forbidden fat*Hoffman**graphs*for G 3 . ...##
###
A relative bound for independence
[article]

2019
*
arXiv
*
pre-print

We prove an upper bound for

arXiv:1901.00585v1
fatcat:u57iaxiggrhhpc6v4aoltumrry
*the*independence*number**of**a**graph*in terms*of**the*largest Laplacian eigenvalue, and*of**a*certain induced subgraph. ... Our bound is*a*refinement*of**a*well-known*Hoffman*-type bound. ... Herein, we focus on eigenvalue bounds for*the*independence*number**of**a**graph*. Let X be*a*non-empty*graph*on n vertices.*The*independence*number**of*X is denoted, as usual, by α. ...##
###
Recent progress on graphs with fixed smallest eigenvalue
[article]

2020
*
arXiv
*
pre-print

Our survey mainly consists

arXiv:2011.11935v1
fatcat:nvdt7xrml5f7xoyrfztogiwega
*of**the*following two parts: (i)*Hoffman**graphs*,*the*basic theory related to*Hoffman**graphs*and*the*applications*of**Hoffman**graphs*to*graphs*with fixed smallest eigenvalue and ... At*the*end*of**the*survey, we also discuss signed*graphs*with fixed smallest eigenvalue and present some new findings. ... Let µ ≤ −1 be*a*real*number*and h*a**Hoffman**graph*with λ min (h) ≥ µ. ...##
###
A generalization of a theorem of Hoffman
[article]

2018
*
arXiv
*
pre-print

For

arXiv:1612.07085v2
fatcat:5nutfx4bb5ev7gh23ot5azcs5i
*the*proof, we use*a*combinatorial object named*Hoffman**graph*, introduced by Woo and Neumaier in 1995. ... In 1977,*Hoffman*gave*a*characterization*of**graphs*with smallest eigenvalue at least -2. In this paper we generalize this result to*graphs*with smaller smallest eigenvalue. ...*A**graph*G is called walk-regular if*the**number**of*closed walks*of*length r starting at*a*given vertex x is independent*of**the*choice*of*x for each r. ...##
###
Fat Hoffman graphs with smallest eigenvalue greater than −3

2014
*
Discrete Applied Mathematics
*

In this paper, we give

doi:10.1016/j.dam.2014.01.008
fatcat:amih4d2bmfaolph6tj32jfk3iq
*a*combinatorial characterization*of**the*special*graphs**of*fat*Hoffman**graphs*containing K_1,2 with smallest eigenvalue greater than -3, where K_1,2 is*the**Hoffman**graph*having one ... Block*graphs**A*vertex v in*a**graph*G is called*a*cut vertex*of*G if*the**number**of*connected components*of*G − v is greater than that*of*G. ... Their results revealed that*graphs*with smallest eigenvalue at least −2 are generalized line*graphs*, except*a*finite*number**of**graphs*represented by*the*root system E 8 . ...##
###
Fat Hoffman graphs with smallest eigenvalue at least −1 − τ

2013
*
Ars Mathematica Contemporanea
*

In this paper, we show that all fat

doi:10.26493/1855-3974.287.137
fatcat:j47ssu7bqfbebobnzd3mvoxtdq
*Hoffman**graphs*with smallest eigenvalue at least −1 − τ , where τ is*the*golden ratio, can be described by*a*finite set*of*fat (−1 − τ )-irreducible*Hoffman**graphs*. ... In*the*terminology*of*Woo and Neumaier, we mean that every fat*Hoffman**graph*with smallest eigenvalue at least −1 − τ is an H-line*graph*, where H is*the*set*of*isomorphism classes*of*maximal fat (−1 − ... Let α be*a*negative real*number*and let H be*a**Hoffman**graph*with λ min (H) ≥ α. ...##
###
Fat Hoffman graphs with smallest eigenvalue at least -1-τ
[article]

2013
*
arXiv
*
pre-print

In this paper, we show that all fat

arXiv:1111.7284v4
fatcat:3tj6wnilxvc6fpdlze2riv3hmy
*Hoffman**graphs*with smallest eigenvalue at least -1-\tau, where \tau is*the*golden ratio, can be described by*a*finite set*of*fat (-1-\tau)-irreducible*Hoffman**graphs*... In*the*terminology*of*Woo and Neumaier, we mean that every fat*Hoffman**graph*with smallest eigenvalue at least -1-\tau is an H-line*graph*, where H is*the*set*of*isomorphism classes*of*maximal fat (-1-\ ...*A*disconnected*Hoffman**graph*is decomposable. Definition 2.11. Let α be*a*negative real*number*. Let H be*a**Hoffman**graph*with λ min (H) ≥ α. ...##
###
On graphs whose smallest eigenvalue is at least − 1 − √2

1995
*
Linear Algebra and its Applications
*

*The*main result is that if

*the*smallest eigenvalue

*of*

*a*

*graph*H exceeds

*a*fixed

*number*larger than

*the*smallest root (= -2.4812)

*of*

*the*polynomial r3 + 231" -2x -2, and if every vertex

*of*H has suffkiently ... large valency, then

*the*smallest eigenvalue

*of*H is at least -1 -fi and

*the*structure

*of*H is completely characterized through

*a*new generalization

*of*line

*graphs*. 1. ... Let 2 be

*a*family

*of*

*Hoffman*

*graphs*. An ZZine

*graph*is

*a*subgraph

*of*

*a*

*graph*H with

*the*following property: THEOREM. ...

##
###
A note on Hoffman-type identities of graphs

2005
*
Linear Algebra and its Applications
*

An eigenvalue

doi:10.1016/j.laa.2004.12.017
fatcat:e5jzsuddzfbd7mr72mhfj7roiu
*of**a**graph*G is called main eigenvalue if it has an eigenvector*the*sum*of*whose entries is not equal to zero.*Hoffman*[A.J.*Hoffman*, On*the*polynomial*of**a**graph*, Amer. Math. ... Monthly 70 (1963) 30-36] proved that G is*a*connected k-regular*graph*if and only if n t i=2 (*A*− λ i I ) = t i=2 (k − λ i ) • J , where I is*the*unit matrix and J*the*all-one matrix and λ 1 = k, λ 2 , ... Acknowledgement This work was completed while*the*first author visited*the*Academy*of*Mathematics and System, Chinese Academy*of*Sciences.*The*authors wish to thank*the*referee for pointing out Ref. ...##
###
The integrally representable trees of norm 3
[article]

2016
*
arXiv
*
pre-print

In this paper, we determine

arXiv:1612.06971v1
fatcat:ahwrfueannazlewfz26mussw6a
*the*integrally representable trees*of*norm 3. ... Note that for help*of*understanding,*the**numbers*in*the**Hoffman**graphs*in Figure 3 and Figure 4 denote*the**number**of*its fat neighbors in*the**Hoffman**graph*h. ... Let µ be*a*real*number*with µ ≤ −1 and let h be*a**Hoffman**graph*with λ min (h) ≥ µ. ...##
###
The integrally representable trees of norm $3$

2017
*
Annals of Mathematical Sciences and Applications
*

For

doi:10.4310/amsa.2017.v2.n2.a7
fatcat:kjbupncmy5bkjic4cxfiajyzxu
*a*3-seedling t with ψ as its reduced representation*of*norm 3, let Λ red (t, 3) be*the*lattice generated by vectors ψ(x) with x ∈ V s (h). Problem 2. ... Is*the*lattice Λ red (t, 3) always 2-integrable for*a*3-seedling t? We do not think so, but we do not have*a*counter example yet. ... Let μ be*a*real*number*with μ ≤ −1 and let h be*a**Hoffman**graph*with λ min (h) ≥ μ. ...##
###
On the limit points of the smallest eigenvalues of regular graphs

2011
*
Designs, Codes and Cryptography
*

Moreover we determine

doi:10.1007/s10623-011-9575-0
fatcat:qwgioqzkp5ftrjclkkohjs3q64
*the*supremum*of**the*smallest eigenvalue among all connected 3-regular*graphs*with smallest eigenvalue less than −2 and we give*the*unique*graph*with this supremum value as its smallest ... From these results, we determine*the*largest and second largest limit points*of*smallest eigenvalues*of*regular*graphs*less than −2. ... Acknowledgements Part*of*this work was done while visiting*the*Graduate School*of*Information Sciences(GSIS) at Tohoku University.*The*author greatly appreciates*the*hospitality*of*Profs. ...
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