scholarly journals An Eigenvalue Characterization of Antipodal Distance-Regular Graphs

10.37236/1315 ◽  
1997 ◽  
Vol 4 (1) ◽  
Author(s):  
M. A. Fiol
Keyword(s):  
Regular Graph ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Graph ◽  
Regular Graphs ◽  
Distance Regular Graph ◽  
Distance Regular Graphs

Let $G$ be a regular (connected) graph with $n$ vertices and $d+1$ distinct eigenvalues. As a main result, it is shown that $G$ is an $r$-antipodal distance-regular graph if and only if the distance graph $G_d$ is constituted by disjoint copies of the complete graph $K_r$, with $r$ satisfying an expression in terms of $n$ and the distinct eigenvalues.

Merging the first and third classes in bipartite distance-regular graphs

2019 ◽  
Vol 12 (07) ◽  
pp. 2050009
Author(s):  
Siwaporn Mamart ◽  
Chalermpong Worawannotai
Keyword(s):  
Regular Graph ◽  
Connected Graph ◽  
Regular Graphs ◽  
Distance Regular Graph ◽  
Original Graph ◽  
Distance Regular Graphs

Merging the first and third classes in a connected graph is the operation of adding edges between all vertices at distance 3 in the original graph while keeping the original edges. We determine when merging the first and third classes in a bipartite distance-regular graph produces a distance-regular graph.

scholarly journals Crooked Functions, Bent Functions, and Distance Regular Graphs

10.37236/1372 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
T. D. Bending ◽  
D. Fon-Der-Flaass
Keyword(s):  
Regular Graph ◽  
Complete Graph ◽  
Bent Functions ◽  
Vector Spaces ◽  
Regular Graphs ◽  
Dimensional Vector ◽  
Distance Regular Graph ◽  
Distance Regular Graphs ◽  
Crooked Functions

Let $V$ and $W$ be $n$-dimensional vector spaces over $GF(2)$. A mapping $Q:V\rightarrow W$ is called crooked if it satisfies the following three properties: $Q(0)=0$; $Q(x)+Q(y)+Q(z)+Q(x+y+z)\neq 0$ for any three distinct $x,y,z$; $Q(x)+Q(y)+Q(z)+Q(x+a)+Q(y+a)+Q(z+a)\neq 0$ if $a\neq 0$ ($x,y,z$ arbitrary). We show that every crooked function gives rise to a distance regular graph of diameter 3 having $\lambda=0$ and $\mu=2$ which is a cover of the complete graph. Our approach is a generalization of a recent construction found by de Caen, Mathon, and Moorhouse. We study graph-theoretical properties of the resulting graphs, including their automorphisms. Also we demonstrate a connection between crooked functions and bent functions.

Some Spectral Characterizations of Strongly Distance-Regular Graphs

2001 ◽  
Vol 10 (2) ◽  
pp. 127-135 ◽  
Author(s):  
M. A. FIOL
Keyword(s):  
Regular Graph ◽  
Connected Graph ◽  
Strongly Regular Graphs ◽  
Regular Graphs ◽  
Distance Regular Graph ◽  
Strongly Regular ◽  
Distance Regular Graphs ◽  
Spectral Characterizations

A graph Γ with diameter d is strongly distance-regular if Γ is distance-regular and its distance-d graph Γd is strongly regular. Some known examples of such graphs are the connected strongly regular graphs, with distance-d graph Γd = Γ (the complement of Γ), and the antipodal distance-regular graphs. Here we study some spectral conditions for a (regular or distance-regular) graph to be strongly distance-regular. In particular, for the case d = 3 the following characterization is proved. A regular (connected) graph Γ, with distinct eigenvalues λ0 > λ1 > λ2 > λ3, is strongly distance-regular if and only if λ2 = −1, and Γ3 is k-regular with degree k satisfying an expression which depends only on the order and the different eigenvalues of Γ.

scholarly journals Thin Distance-Regular Graphs with Classical Parameters $(D,q,q, \frac{q^{t}-1}{q-1}-1)$ with $t> D$ are the Grassmann Graphs

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Ying Ying Tan ◽  
Xiaoye Liang ◽  
Jack Koolen
Keyword(s):  
Regular Graph ◽  
Regular Graphs ◽  
Distance Regular Graph ◽  
Intersection Numbers ◽  
Survey Paper ◽  
Grassmann Graph ◽  
Distance Regular Graphs ◽  
Classical Parameters

In the survey paper by Van Dam, Koolen and Tanaka (2016), they asked to classify the thin $Q$-polynomial distance-regular graphs. In this paper, we show that a thin distance-regular graph with the same intersection numbers as a Grassmann graph $J_q(n, D)~ (n \geqslant 2D)$ is the Grassmann graph if $D$ is large enough.

scholarly journals A Short Proof of the Odd-Girth Theorem

10.37236/2289 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Edwin R. Van Dam ◽  
Miquel Angel Fiol
Keyword(s):  
Connected Graph ◽  
Short Proof ◽  
Direct Proof ◽  
Regular Graphs ◽  
Distance Regular Graphs

Recently, it has been shown that a connected graph $\Gamma$ with $d+1$ distinct eigenvalues and odd-girth $2d+1$ is distance-regular. The proof of this result was based on the spectral excess theorem. In this note we present an alternative and more direct proof which does not rely on the spectral excess theorem, but on a known characterization of distance regular graphs in terms of the predistance polynomial of degree $d$.

scholarly journals On Bipartite $Q$-Polynomial Distance-Regular Graphs with Diameter 9, 10, or 11

10.37236/7347 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Štefko Miklavič
Keyword(s):  
Regular Graph ◽  
Regular Graphs ◽  
Distance Regular Graph ◽  
Intersection Numbers ◽  
Distance Regular Graphs

Let $\Gamma$ denote a bipartite distance-regular graph with diameter $D$. In [Caughman (2004)], Caughman showed that if $D \ge 12$, then $\Gamma$ is $Q$-polynomial if and only if one of the following (i)-(iv) holds: (i) $\Gamma$ is the ordinary $2D$-cycle, (ii) $\Gamma$ is the Hamming cube $H(D,2)$, (iii) $\Gamma$ is the antipodal quotient of the Hamming cube $H(2D,2)$, (iv) the intersection numbers of $\Gamma$ satisfy $c_i = (q^i - 1)/(q-1)$, $b_i = (q^D-q^i)/(q-1)$ $(0 \le i \le D)$, where $q$ is an integer at least $2$. In this paper we show that the above result is true also for bipartite distance-regular graphs with $D \in \{9,10,11\}$.

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 On Bipartite $Q$-Polynomial Distance-Regular Graphs with $c_2 \le 2$

10.37236/4556 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Stefko Miklavic ◽  
Safet Penjic
Keyword(s):  
Regular Graph ◽  
Intersection Number ◽  
Regular Graphs ◽  
Distance Regular Graph ◽  
Distance Regular Graphs

Let $\Gamma$ denote a bipartite $Q$-polynomial distance-regular graph with diameter $D \ge 4$, valency $k \ge 3$ and intersection number $c_2 \le 2$. We show that $\Gamma$ is either the $D$-dimensional hypercube, or the antipodal quotient of the $2D$-dimensional hypercube, or $D=5$.

scholarly journals Distance-Regular Graphs with a Relatively Small Eigenvalue Multiplicity

10.37236/2410 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Jack H. Koolen ◽  
Joohyung Kim ◽  
Jongyook Park
Keyword(s):  
Regular Graph ◽  
Small Eigenvalue ◽  
Regular Graphs ◽  
Distance Regular Graph ◽  
Distance Regular Graphs

Godsil showed that if $\Gamma$ is a distance-regular graph with diameter $D \geq 3$ and valency $k \geq 3$, and $\theta$ is an eigenvalue of $\Gamma$ with multiplicity $m \geq 2$, then $k \leq\frac{(m+2)(m-1)}{2}$.In this paper we will give a refined statement of this result. We show that if $\Gamma$ is a distance-regular graph with diameter $D \geq 3$, valency $k \geq 2$ and an eigenvalue $\theta$ with multiplicity $m\geq 2$, such that $k$ is close to $\frac{(m+2)(m-1)}{2}$, then $\theta$ must be a tail. We also characterize the distance-regular graphs with diameter $D \geq 3$, valency $k \geq 3$ and an eigenvalue $\theta$ with multiplicity $m \geq 2$ satisfying $k= \frac{(m+2)(m-1)}{2}$.

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.

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