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Edge-distance-regular graphs [PDF]
Journal of Combinatorial Theory, Series A, 2011 Edge-distance-regularity is a concept recently introduced by the authors which is similar to that of distance-regularity, but now the graph is seen from each of its edges instead of from its vertices. More precisely, a graph Γ with adjacency matrix A is edge-distance-regular when it is distance-regular around each of its edges and with the same ...
Cámara Vallejo, Marc +4 more
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ON DISTANCE–REGULAR GRAPHS OF DIAMETER 3 WITH EIGENVALUE \(\theta=1\)
Ural Mathematical Journal, 2022 For a distance-regular graph \(\Gamma\) of diameter 3, the graph \(\Gamma_i\) can be strongly regular for \(i=2\) or 3. J.Kulen and co-authors found the parameters of a strongly regular graph \(\Gamma_2\) given the intersection array of the graph ...
Alexander A. Makhnev +2 more
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Al-Rafidain Journal of Computer Sciences and Mathematics, 2023 The new distance defined on a connected graph G contains of three terms: The ordinary distance between any two vertices in G, both the sum and the product of the two vertices' degrees, as this distance is more useful than the ordinary distance ...
Asmaa Aziz
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Two distance-regular graphs [PDF]
Journal of Algebraic Combinatorics, 2011 We construct two families of distance-regular graphs, namely the subgraph of the dual polar graph of type B_3(q) induced on the vertices far from a fixed point, and the subgraph of the dual polar graph of type D_4(q) induced on the vertices far from a fixed edge. The latter is the extended bipartite double of the former.
Brouwer, Andries E. +1 more
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Transitive distance-regular graphs from linear groups $L(3,q)$, $q = 2,3,4,5$ [PDF]
Transactions on Combinatorics, 2020 In this paper we classify distance-regular graphs, including strongly regular graphs, admitting a transitive action of the linear groups $L(3,2)$, $L(3,3)$, $L(3,4)$ and $L(3,5)$ for which the rank of the permutation representation is at most 15.
Andrea Svob
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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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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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DISTANCE-REGULAR GRAPH WITH INTERSECTION ARRAY {27, 20, 7; 1, 4, 21} DOES NOT EXIST
Ural Mathematical Journal, 2020 In the class of distance-regular graphs of diameter 3 there are 5 intersection arrays of graphs with at most 28 vertices and noninteger eigenvalue. These arrays are \(\{18,14,5;1,2,14\}\), \(\{18,15,9;1,1,10\}\), \(\{21,16,10;1,2,12\}\), \(\{24,21,3;1,3 ...
Konstantin S. Efimov +1 more
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NEW BOUNDS AND EXTREMAL GRAPHS FOR DISTANCE SIGNLESS LAPLACIAN SPECTRAL RADIUS [PDF]
Journal of Algebraic Systems, 2021 The distance signless Laplacian spectral radius of a connected graph $G$ is the largest eigenvalue of the distance signless Laplacian matrix of $G$, defined as $D^{Q}(G)=Tr(G)+D(G)$, where $D(G)$ is the distance matrix of $G$ and $Tr(G)$ is the diagonal ...
A. Alhevaz, M. Baghipur, S. Paul
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