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

Pawel Wocjan and Clive Elphick and David Anekstein
2020 arXiv   pre-print
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.  ...

### Sesqui-regular graphs with fixed smallest eigenvalue [article]

Jack H. Koolen, Brhane Gebremichel, Jae Young Yang, Qianqian Yang
2021 arXiv   pre-print
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.  ...

### On the order of regular graphs with fixed second largest eigenvalue [article]

Jae Young Yang, Jack H. Koolen
2018 arXiv   pre-print
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.  ...

### On graphs with smallest eigenvalue at least -3 and their lattices [article]

Jack H. Koolen, Jae Young Yang, Qianqian Yang
2018 arXiv   pre-print
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 .  ...

### A relative bound for independence [article]

Bogdan Nica
2019 arXiv   pre-print
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.  ...

### Recent progress on graphs with fixed smallest eigenvalue [article]

Jack H. Koolen, Meng-Yue Cao, Qianqian Yang
2020 arXiv   pre-print
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.  ...

### A generalization of a theorem of Hoffman [article]

Jack H. Koolen, Qianqian Yang, Jae Young Yang
2018 arXiv   pre-print
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.  ...

### Fat Hoffman graphs with smallest eigenvalue greater than −3

Akihiro Munemasa, Yoshio Sano, Tetsuji Taniguchi
2014 Discrete Applied Mathematics
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 .  ...

### Fat Hoffman graphs with smallest eigenvalue at least −1 − τ

Akihiro Munemasa, Yoshio Sano, Tetsuji Taniguchi
2013 Ars Mathematica Contemporanea
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.  ...

### Fat Hoffman graphs with smallest eigenvalue at least -1-τ [article]

Akihiro Munemasa, Yoshio Sano, Tetsuji Taniguchi
2013 arXiv   pre-print
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.  ...

### On graphs whose smallest eigenvalue is at least − 1 − √2

Renee Woo, Arnold Neumaier
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

Yaoping Hou, Feng Tian
2005 Linear Algebra and its Applications
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.  ...

### The integrally representable trees of norm 3 [article]

Jack H. Koolen, Masood Ur Rehman, Qianqian Yang
2016 arXiv   pre-print
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.  ...

### The integrally representable trees of norm $3$

Jack H. Koolen, Masood Ur Rehman, Qianqian Yang
2017 Annals of Mathematical Sciences and Applications
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.  ...

### On the limit points of the smallest eigenvalues of regular graphs

Hyonju Yu
2011 Designs, Codes and Cryptography
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.  ...
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