The Terwilliger algebra of an almost-bipartite distance-regular graph and its antipodal 2-cover
Discrete Mathematics, 2000 B. Collins
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The Terwilliger Algebra of a 2-Homogeneous Bipartite Distance-Regular Graph
Journal of Combinatorial Theory Series B, 2001 B. Curtin
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The Terwilliger Algebra Associated with a Set of Vertices in a Distance-Regular Graph
Journal of Algebraic Combinatorics, 2005 H. Suzuki
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SHILLA GRAPHS WITH \(b=5\) AND \(b=6\)
Ural Mathematical Journal, 2021 A \(Q\)-polynomial Shilla graph with \(b = 5\) has intersection arrays \(\{105t,4(21t+1),16(t+1); 1,4 (t+1),84t\}\), \(t\in\{3,4,19\}\). The paper proves that distance-regular graphs with these intersection arrays do not exist.
Alexander A. Makhnev, Ivan N. Belousov
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Tight Distance-Regular Graphs [PDF]
Journal of Algebraic Combinatorics, 2000 We consider a distance-regular graph $\G$ with diameter $d \ge 3$ and eigenvalues $k= _0> _1>... > _d$. We show the intersection numbers $a_1, b_1$ satisfy $$ ( _1 + {k \over a_1+1}) ( _d + {k \over a_1+1}) \ge - {ka_1b_1 \over (a_1+1)^2}. $$ We say $\G$ is {\it tight} whenever $\G$ is not bipartite, and equality holds above.
Jurišić, Aleksandar +2 more
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The uniqueness of a distance-regular graph with intersection array $$\{32,27,8,1;1,4,27,32\}$${32,27,8,1;1,4,27,32} and related results [PDF]
Des. Codes Cryptogr., 2015 It is known that, up to isomorphism, there is a unique distance-regular graph $$\Delta $$Δ with intersection array $$\{32,27;1,12\}$${32,27;1,12} [equivalently, $$\Delta $$Δ is the unique strongly regular graph with parameters (105, 32, 4, 12)].
L. H. Soicher
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An inequality involving two eigenvalues of a bipartite distance-regular graph
Discrete Mathematics, 2000 Mark S. MacLean
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The Generalized Distance Spectrum of the Join of Graphs [PDF]
, 2020 Let G be a simple connected graph. In this paper, we study the spectral properties of the generalized distance matrix of graphs, the convex combination of the symmetric distance matrix D(G) and diagonal matrix of the vertex transmissions Tr(G) .
Alhevaz, Abdollah +3 more
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ON A CLASS OF EDGE-TRANSITIVE DISTANCE-REGULAR ANTIPODAL COVERS OF COMPLETE GRAPHS
Ural Mathematical Journal, 2021 The paper is devoted to the problem of classification of edge-transitive distance-regular antipodal covers of complete graphs. This extends the classification of those covers that are arc-transitive, which has been settled except for some tricky cases ...
Ludmila Yu. Tsiovkina
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The Terwilliger polynomial of a Q-polynomial distance-regular graph and its application to the pseudo-partition graphs [PDF]
, 2014 Let $\Gamma$ be a $Q$-polynomial distance-regular graph with diameter at least $3$. Terwilliger (1993) implicitly showed that there exists a polynomial, say $T(\lambda)\in \mathbb{C}[\lambda]$, of degree $4$ depending only on the intersection numbers of $
Alexander L. Gavrilyuk, J. Koolen
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