On nonexistence of distance regular graphs with the intersection array ${53,40,28,16;1,4,10,28}$

2021 ◽  
Vol 28 (3) ◽  
pp. 38-48
Author(s):  
A. A. Makhnev ◽  
M. P. Golubyatnikov
Keyword(s):  
Intersection Array ◽  
Regular Graphs ◽  
Distance Regular Graphs

scholarly journals The Spectral Excess Theorem for Distance-Biregular Graphs.

10.37236/3305 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Miquel Àngel Fiol
Keyword(s):  
Bipartite Graph ◽  
Connected Graph ◽  
Intersection Array ◽  
Stable Set ◽  
Regular Graphs ◽  
Distance Regular Graphs

The spectral excess theorem for distance-regular graphs states that a regular (connected) graph is distance-regular if and only if its spectral-excess equals its average excess. A bipartite graph $\Gamma$ is distance-biregular when it is distance-regular around each vertex and the intersection array only depends on the stable set such a vertex belongs to. In this note we derive a new version of the spectral excess theorem for bipartite distance-biregular graphs.

scholarly journals Using Symbolic Computation to Prove Nonexistence of Distance-Regular Graphs

10.37236/7763 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Janoš Vidali
Keyword(s):  
Symbolic Computation ◽  
Computer Algebra ◽  
Regular Graph ◽  
Computer Algebra System ◽  
Intersection Array ◽  
Regular Graphs ◽  
Distance Regular Graph ◽  
Triple Intersection ◽  
Distance Regular Graphs ◽  
Krein Condition

A package for the Sage computer algebra system is developed for checking feasibility of a given intersection array for a distance-regular graph. We use this tool to show that there is no distance-regular graph with intersection array$$\{(2r+1)(4r+1)(4t-1), 8r(4rt-r+2t), (r+t)(4r+1); 1, (r+t)(4r+1), 4r(2r+1)(4t-1)\}  (r, t \geq 1),$$$\{135,\! 128,\! 16; 1,\! 16,\! 120\}$, $\{234,\! 165,\! 12; 1,\! 30,\! 198\}$ or $\{55,\! 54,\! 50,\! 35,\! 10; 1,\! 5,\! 20,\! 45,\! 55\}$. In all cases, the proofs rely on equality in the Krein condition, from which triple intersection numbers are determined. Further combinatorial arguments are then used to derive nonexistence. 

scholarly journals DISTANCE-REGULAR GRAPH WITH INTERSECTION ARRAY {27, 20, 7; 1, 4, 21} DOES NOT EXIST

2020 ◽  
Vol 6 (2) ◽  
pp. 63
Author(s):  
Konstantin S. Efimov ◽  
Alexander A. Makhnev
Keyword(s):  
Regular Graph ◽  
Intersection Array ◽  
Regular Graphs ◽  
Distance Regular Graph ◽  
Automorphisms Of Graphs ◽  
Distance Regular Graphs

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 \(\{18,14,5;1,2,14\}\), \(\{18,15,9;1,1,10\}\), \(\{21,16,10;1,2,12\}\), \(\{24,21,3;1,3,18\}\), and \(\{27,20,7;1,4,21\}\). Automorphisms of graphs with intersection arrays \(\{18,15,9;1,1,10\}\) and \(\{24,21,3;1,3,18\}\) were found earlier by A.A. Makhnev and D.V. Paduchikh. In this paper, it is proved that a graph with the intersection array \(\{27,20,7;1,4,21\}\) does not exist.

scholarly journals The matching polynomial of a distance-regular graph

2000 ◽  
Vol 23 (2) ◽  
pp. 89-97 ◽  
Author(s):  
Robert A. Beezer ◽  
E. J. Farrell
Keyword(s):  
Regular Graph ◽  
Small Diameter ◽  
Intersection Array ◽  
Regular Graphs ◽  
Distance Regular Graph ◽  
Intersection Numbers ◽  
Matching Polynomial ◽  
Distance Regular Graphs

A distance-regular graph of diameterdhas2dintersection numbers that determine many properties of graph (e.g., its spectrum). We show that the first six coefficients of the matching polynomial of a distance-regular graph can also be determined from its intersection array, and that this is the maximum number of coefficients so determined. Also, the converse is true for distance-regular graphs of small diameter—that is, the intersection array of a distance-regular graph of diameter 3 or less can be determined from the matching polynomial of the graph.

scholarly journals On the spectra of certain distance-regular graphs

1979 ◽  
Vol 27 (3) ◽  
pp. 274-293 ◽  
Author(s):  
Eiichi Bannai ◽  
Tatsuro Ito
Keyword(s):  
Regular Graphs ◽  
Distance Regular Graphs
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