Triangle-free distance-regular graphs with an eigenvalue multiplicity equal to their valency and diameter 3

Jurišić, Aleksandar; Koolen, Jack H.; itnik, Arjana

doi:10.1016/j.ejc.2006.10.001

Cited by 4 publications

(2 citation statements)

References 9 publications

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“…For µ = 1, this intersection array is uniquely realized by the dodecahedron. Jurišić, Koolen, and Žitnik [383] showed among other results that if Γ is primitive and has an eigenvalue with multiplicity k, a 1 = 0, and D = 3, then the association scheme underlying Γ is formally self-dual and thus Γ is Q-polynomial and 1-homogeneous.…”

Section: More Results On Homogeneitymentioning

confidence: 98%

Distance-regular graphs

van Dam,

Koolen,

Tanaka

2014

Preprint

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Section: More Results On Homogeneitymentioning

confidence: 98%

Distance-regular graphs

van Dam,

Koolen,

Tanaka

2014

Preprint

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“…Yamazaki [30] proved that if a bipartite distance-regular graph has an eigenvalue with multiplicity equals its valency, then such graph is 2-homogeneous. Furthermore, the relationship between the eigenvalue multiplicity and the valency of triangle-free distance-regular graphs was investigated in [6,11,12]. The upper bounds on the eigenvalue multiplicity for cubic graphs and triangle-free graphs were considered in [24,25].…”

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confidence: 99%

The Multiplicity of $A_{\alpha}$-eigenvalues of Graphs

Xue

Liu

et al. 2020

ELA

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For a graph $G$ and real number $\alpha\in [0,1]$, the $A_{\alpha}$-matrix of $G$ is defined as $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G),$ where $A(G)$ is the adjacency matrix of $G$ and $D(G)$ is the diagonal matrix of the vertex degrees of $G$. In this paper, the largest multiplicity of the $A_{\alpha}$-eigenvalues of a broom tree is considered, and all graphs with an $A_{\alpha}$-eigenvalue of multiplicity at least $n-2$ are characterized.

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On triangle-free distance-regular graphs with an eigenvalue multiplicity equal to the valency

Coolsaet

Jurišić

Koolen

2008

European Journal of Combinatorics

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Let Gamma be a triangle-free distance-regular graph with diameter d >= 3, valency k >= 3 and intersection number a(2) not equal 0. Assume Gamma has an eigenvalue with multiplicity k. We show that Gamma is 1-homogeneous in the sense of Nomura when d = 3 or when d >= 4 and a(4) = 0. In the latter case we prove that r is an antipodal cover of a strongly regular graph, which means that it has diameter 4 or 5. For d = 5 the following infinite family of feasible intersection arrays: {2 mu(2) + mu, 2 mu(2) + mu -1, mu(2), mu,1; 1, mu, mu(2), 2 mu(2) + mu - 1, 2 mu(2) + mu}, mu is an element of N, is known. For mu = 1 the intersection array is uniquely realized by the dodecahedron. For mu = 1 we show that there are no distance-regular graphs with this intersection array

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Triangle-free distance-regular graphs with an eigenvalue multiplicity equal to their valency and diameter 3

Cited by 4 publications

References 9 publications

Distance-regular graphs

Distance-regular graphs

The Multiplicity of $A_{\alpha}$-eigenvalues of Graphs

On triangle-free distance-regular graphs with an eigenvalue multiplicity equal to the valency

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