scholarly journals An Application of Hoffman Graphs for Spectral Characterizations of Graphs

10.37236/6428 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Qianqian Yang ◽  
Aida Abiad ◽  
Jack H. Koolen
Keyword(s):  
Regular Graph ◽  
Distance Regular Graph ◽  
Intersection Numbers ◽  
Grassmann Graph ◽  
Spectral Characterizations

In this paper, we present the first application of Hoffman graphs for spectral characterizations of graphs. In particular, we show that the 2-clique extension of the $(t+1)\times (t+1)$-grid is determined by its spectrum when $t$ is large enough. This result will help to show that the Grassmann graph $J_2(2D,D)$ is determined by its intersection numbers as a distance regular graph, if $D$ is large enough.

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 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 Equivalent Characterizations of the Spectra of Graphs and Applications to Measures of Distance-regularity

2020 ◽  
Vol 36 (36) ◽  
pp. 629-644
Author(s):  
Miquel Àngel Fiol ◽  
Josep Fàbrega ◽  
Victor Diego
Keyword(s):  
Regular Graph ◽  
Maximum Distance ◽  
Combinatorial Structure ◽  
Distance Regular Graph ◽  
Intersection Numbers ◽  
Combinatorial Properties ◽  
Spectra Of Graphs ◽  
A Value ◽  
The Mean ◽  
The Given

The spectrum of a graph usually provides a lot of information about its combinatorial structure. Moreover, from the spectrum, the so-called predistance polynomials can be defined, as a generalization, for any graph, of the distance polynomials of a distance-regular graph. Going further, the preintersection numbers generalize the intersection numbers of a distance-regular graph. This paper describes, for any graph, the closed relationships between its spectrum, predistance polynomials, and preintersection numbers. Then, some applications to derive combinatorial properties of the given graph, most of them related to some fundamental characterizations of distance-regularity, are presented. In particular, the so-called `spectral excess theorem' is revisited. This result states that a connected regular graph is distance-regular if and only if its spectral excess, which is a value computed from the spectrum, equals the average excess, that is, the mean of the numbers of vertices at maximum distance from every vertex.

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.

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 Nonexistence of a Class of Distance-Regular Graphs

10.37236/3356 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Yu-pei Huang ◽  
Yeh-jong Pan ◽  
Chih-wen Weng
Keyword(s):  
Regular Graph ◽  
Regular Graphs ◽  
Distance Regular Graph ◽  
Intersection Numbers ◽  
Distance Regular Graphs ◽  
Classical Parameters ◽  
Alpha Beta ◽  
Beta 2

Let $\Gamma$ denote a distance-regular graph with diameter $D \geq 3$ and intersection numbers $a_1=0, a_2 \neq 0$, and $c_2=1$. We show a connection between the $d$-bounded property and the nonexistence of parallelograms of any length up to $d+1$. Assume further that $\Gamma$ is with classical parameters $(D, b, \alpha, \beta)$, Pan and Weng (2009) showed that $(b, \alpha, \beta)= (-2, -2, ((-2)^{D+1}-1)/3).$ Under the assumption $D \geq 4$, we exclude this class of graphs by an application of the above connection.

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