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More Tales of Hoffman: bounds for the vector chromatic number of a graph
[article]
2020
arXiv
pre-print
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Let χ(G) denote 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. ...
arXiv:1812.02613v3
fatcat:es5rf2iupnetzh4mfdqckna524
Sesqui-regular graphs with fixed smallest eigenvalue
[article]
2021
arXiv
pre-print
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In this paper, we will study 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. ...
arXiv:1904.01274v2
fatcat:n3ewqfq5gza2nns2b6smhfboti
On the order of regular graphs with fixed second largest eigenvalue
[article]
2018
arXiv
pre-print
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Let v(k, λ) be 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. ...
arXiv:1809.01888v1
fatcat:uggmkos7nzggtnl226fjyb2nfi
On graphs with smallest eigenvalue at least -3 and their lattices
[article]
2018
arXiv
pre-print
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This result generalizes 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 . ...
arXiv:1804.00369v1
fatcat:dvvlmudxsbcaneappmk2fbsrse
A relative bound for independence
[article]
2019
arXiv
pre-print
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We prove an upper bound for 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. ...
arXiv:1901.00585v1
fatcat:u57iaxiggrhhpc6v4aoltumrry
Recent progress on graphs with fixed smallest eigenvalue
[article]
2020
arXiv
pre-print
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Our survey mainly consists 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. ...
arXiv:2011.11935v1
fatcat:nvdt7xrml5f7xoyrfztogiwega
A generalization of a theorem of Hoffman
[article]
2018
arXiv
pre-print
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For 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. ...
arXiv:1612.07085v2
fatcat:5nutfx4bb5ev7gh23ot5azcs5i
Fat Hoffman graphs with smallest eigenvalue greater than −3
2014
Discrete Applied Mathematics
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2013 pre-print arXiv:1211.3929v3 fatcat:3zbst5w4ybb6pbihlztvme5itiOther Versions
In this paper, we give 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 . ...
doi:10.1016/j.dam.2014.01.008
fatcat:amih4d2bmfaolph6tj32jfk3iq
Fat Hoffman graphs with smallest eigenvalue at least −1 − τ
2013
Ars Mathematica Contemporanea
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In this paper, we show that all fat 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. ...
doi:10.26493/1855-3974.287.137
fatcat:j47ssu7bqfbebobnzd3mvoxtdq
Fat Hoffman graphs with smallest eigenvalue at least -1-τ
[article]
2013
arXiv
pre-print
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In this paper, we show that all fat 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. ...
arXiv:1111.7284v4
fatcat:3tj6wnilxvc6fpdlze2riv3hmy
On graphs whose smallest eigenvalue is at least − 1 − √2
1995
Linear Algebra and its Applications
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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. ...
doi:10.1016/0024-3795(95)00245-m
fatcat:6n5wdnlsavfarpwnvk3a4t563a
A note on Hoffman-type identities of graphs
2005
Linear Algebra and its Applications
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An eigenvalue 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. ...
doi:10.1016/j.laa.2004.12.017
fatcat:e5jzsuddzfbd7mr72mhfj7roiu
The integrally representable trees of norm 3
[article]
2016
arXiv
pre-print
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In this paper, we determine 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. ...
arXiv:1612.06971v1
fatcat:ahwrfueannazlewfz26mussw6a
The integrally representable trees of norm $3$
2017
Annals of Mathematical Sciences and Applications
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For 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. ...
doi:10.4310/amsa.2017.v2.n2.a7
fatcat:kjbupncmy5bkjic4cxfiajyzxu
On the limit points of the smallest eigenvalues of regular graphs
2011
Designs, Codes and Cryptography
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Moreover we determine 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. ...
doi:10.1007/s10623-011-9575-0
fatcat:qwgioqzkp5ftrjclkkohjs3q64
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