scholarly journals On the Clique Number of a Strongly Regular Graph

10.37236/7873 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Gary R. W. Greaves ◽  
Leonard H. Soicher
Keyword(s):  
Regular Graph ◽  
Strongly Regular Graph ◽  
Clique Number ◽  
Upper Bounds ◽  
Strongly Regular Graphs ◽  
Regular Graphs ◽  
Strongly Regular ◽  
Paley Graphs

We determine new upper bounds for the clique numbers of strongly regular graphs in terms of their parameters. These bounds improve on the Delsarte bound for infinitely many feasible parameter tuples for strongly regular graphs, including infinitely many parameter tuples that correspond to Paley graphs.

scholarly journals On strongly regular graphs with m2 = qm3 and m3 = qm2 where q ∈ Q

2021 ◽  
Vol 109 (123) ◽  
pp. 35-60
Author(s):  
Mirko Lepovic
Keyword(s):  
Regular Graph ◽  
Complete Graph ◽  
Strongly Regular Graph ◽  
Strongly Regular Graphs ◽  
Regular Graphs ◽  
Strongly Regular

We say that a regular graph G of order n and degree r ? 1 (which is not the complete graph) is strongly regular if there exist non-negative integers ? and ? such that |Si ? Sj | = ? for any two adjacent vertices i and j, and |Si ? Sj | = ? for any two distinct non-adjacent vertices i and j, where Sk denotes the neighborhood of the vertex k. Let ?1 = r, ?2 and ?3 be the distinct eigenvalues of a connected strongly regular graph. Let m1 = 1, m2 and m3 denote the multiplicity of r, ?2 and ?3, respectively. We here describe the parameters n, r, ? and ? for strongly regular graphs with m2 = qm3 and m3 = qm2 for q = 3/2, 4/3, 5/2, 5/3, 5/4, 6/5.

scholarly journals An example of using star complements in classifying strongly regular graphs

Filomat ◽  
2008 ◽  
Vol 22 (2) ◽  
pp. 53-57 ◽  
Author(s):  
Marko Milosevic
Keyword(s):  
Regular Graph ◽  
Strongly Regular Graph ◽  
Strongly Regular Graphs ◽  
Regular Graphs ◽  
Strongly Regular

In this paper we show how the star complement technique can be used to reprove the result of Wilbrink and Brouwer that the strongly regular graph with parameters (57, 14, 1, 4) does not exist. .

scholarly journals Binary Representations of Regular Graphs

2011 ◽  
Vol 2011 ◽  
pp. 1-15
Author(s):  
Yury J. Ionin
Keyword(s):  
Regular Graph ◽  
Line Graph ◽  
Strongly Regular Graph ◽  
Regular Graphs ◽  
Spherical Representation ◽  
Least Eigenvalue ◽  
Vertex Set ◽  
Strongly Regular ◽  
Distance Set ◽  
Hamming Space

For any 2-distance set in the n-dimensional binary Hamming space , let be the graph with as the vertex set and with two vertices adjacent if and only if the distance between them is the smaller of the two nonzero distances in . The binary spherical representation number of a graph , or bsr(), is the least n such that is isomorphic to , where is a 2-distance set lying on a sphere in . It is shown that if is a connected regular graph, then bsr, where b is the order of and m is the multiplicity of the least eigenvalue of , and the case of equality is characterized. In particular, if is a connected strongly regular graph, then bsr if and only if is the block graph of a quasisymmetric 2-design. It is also shown that if a connected regular graph is cospectral with a line graph and has the same binary spherical representation number as this line graph, then it is a line graph.

scholarly journals Upper bounds on Q-spectral radius of book-free and/or $K_{s,t}$-free graphs

2017 ◽  
Vol 32 ◽  
pp. 447-453
Author(s):  
Qi Kong ◽  
Ligong Wang
Keyword(s):  
Spectral Radius ◽  
Regular Graph ◽  
Maximum Degree ◽  
Strongly Regular Graph ◽  
Upper Bounds ◽  
Free Graph ◽  
Laplacian Spectral Radius ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian ◽  
Strongly Regular

In this paper, we prove two results about the signless Laplacian spectral radius $q(G)$ of a graph $G$ of order $n$ with maximum degree $\Delta$. Let $B_{n}=K_{2}+\overline{K_{n}}$ denote a book, i.e., the graph $B_{n}$ consists of $n$ triangles sharing an edge. The results are the following: (1) Let $1< k\leq l< \Delta < n$ and $G$ be a connected \{$B_{k+1},K_{2,l+1}$\}-free graph of order $n$ with maximum degree $\Delta$. Then $$\displaystyle q(G)\leq \frac{1}{4}[3\Delta+k-2l+1+\sqrt{(3\Delta+k-2l+1)^{2}+16l(\Delta+n-1)}$$ with equality if and only if $G$ is a strongly regular graph with parameters ($\Delta$, $k$, $l$). (2) Let $s\geq t\geq 3$, and let $G$ be a connected $K_{s,t}$-free graph of order $n$ $(n\geq s+t)$. Then $$q(G)\leq n+(s-t+1)^{1/t}n^{1-1/t}+(t-1)(n-1)^{1-3/t}+t-3.$$

Strongly Regular Graphs Derived from Combinatorial Designs

1970 ◽  
Vol 22 (3) ◽  
pp. 597-614 ◽  
Author(s):  
J. M. Goethals ◽  
J. J. Seidel
Keyword(s):  
Strongly Regular Graph ◽  
Hadamard Matrices ◽  
Latin Squares ◽  
Discrete Mathematics ◽  
Strongly Regular Graphs ◽  
Block Designs ◽  
Regular Graphs ◽  
Geometric Configurations ◽  
Strongly Regular ◽  
Definition Of

Several concepts in discrete mathematics such as block designs, Latin squares, Hadamard matrices, tactical configurations, errorcorrecting codes, geometric configurations, finite groups, and graphs are by no means independent. Combinations of these notions may serve the development of any one of them, and sometimes reveal hidden interrelations. In the present paper a central role in this respect is played by the notion of strongly regular graph, the definition of which is recalled below.In § 2, a fibre-type construction for graphs is given which, applied to block designs withλ= 1 and Hadamard matrices, yields strongly regular graphs. The method, although still limited in its applications, may serve further developments. In § 3 we deal with block designs, first considered by Shrikhande[22],in which the number of points in the intersection of any pair of blocks attains only two values.

scholarly journals A Prolific Construction of Strongly Regular Graphs with the $n$-e.c. Property

10.37236/1647 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Peter J. Cameron ◽  
Dudley Stark
Keyword(s):  
Regular Graph ◽  
Prime Power ◽  
Probabilistic Methods ◽  
Strongly Regular Graphs ◽  
Regular Graphs ◽  
Existentially Closed ◽  
Large N ◽  
Vertex Set ◽  
Strongly Regular ◽  
Hadamard Designs

A graph is $n$-e.c.$\,$ ($n$-existentially closed) if for every pair of subsets $U$, $W$ of the vertex set $V$ of the graph such that $U\cap W=\emptyset$ and $|U|+|W|=n$, there is a vertex $v\in V-(U\cup W)$ such that all edges between $v$ and $U$ are present and no edges between $v$ and $W$ are present. A graph is strongly regular if it is a regular graph such that the number of vertices mutually adjacent to a pair of vertices $v_1,v_2\in V$ depends only on whether or not $\{v_1,v_2\}$ is an edge in the graph. The only strongly regular graphs that are known to be $n$-e.c. for large $n$ are the Paley graphs. Recently D. G. Fon-Der-Flaass has found prolific constructions of strongly regular graphs using affine designs. He notes that some of these constructions were also studied by Wallis. By taking the affine designs to be Hadamard designs obtained from Paley tournaments, we use probabilistic methods to show that many non-isomorphic strongly regular $n$-e.c. graphs of order $(q+1)^2$ exist whenever $q\geq 16 n^2 2^{2n}$ is a prime power such that $q\equiv 3\!\!\!\pmod{4}$.

scholarly journals Hadamard Matrices and Strongly Regular Graphs with the $3$-e.c. Adjacency Property

10.37236/1545 ◽  
2000 ◽  
Vol 8 (1) ◽  
Author(s):  
Anthony Bonato ◽  
W. H. Holzmann ◽  
Hadi Kharaghani
Keyword(s):  
Hadamard Matrix ◽  
Infinite Family ◽  
Hadamard Matrices ◽  
Strongly Regular Graphs ◽  
Regular Graphs ◽  
Element Subset ◽  
Strongly Regular ◽  
Almost All ◽  
Paley Graphs

A graph is $3$-e.c. if for every $3$-element subset $S$ of the vertices, and for every subset $T$ of $S$, there is a vertex not in $S$ which is joined to every vertex in $T$ and to no vertex in $S\setminus T$. Although almost all graphs are $3$-e.c., the only known examples of strongly regular $3$-e.c. graphs are Paley graphs with at least $29$ vertices. We construct a new infinite family of $3$-e.c. graphs, based on certain Hadamard matrices, that are strongly regular but not Paley graphs. Specifically, we show that Bush-type Hadamard matrices of order $16n^2$ give rise to strongly regular $3$-e.c. graphs, for each odd $n$ for which $4n$ is the order of a Hadamard matrix.

scholarly journals Bounds for the Hückel Energy of a Graph

10.37236/223 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Ebrahim Ghorbani ◽  
Jack H. Koolen ◽  
Jae Young Yang
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Electron Energy ◽  
Regular Graph ◽  
Strongly Regular Graph ◽  
Infinite Family ◽  
Upper Bounds ◽  
Molecular Graphs ◽  
Strongly Regular

Let $G$ be a graph on $n$ vertices with $r := \lfloor n/2 \rfloor$ and let $\lambda _1 \geq\cdots\geq \lambda _{n} $ be adjacency eigenvalues of $G$. Then the Hückel energy of $G$, HE($G$), is defined as $${\rm HE}(G) = \cases{ \displaystyle \; 2\sum_{i=1}^{r} \lambda_i, & \hbox{if $n= 2r$;} \cr \displaystyle \; 2\sum_{i=1}^{\phantom{l}r\phantom{l}} \lambda_i + \lambda_{r+1}, & \hbox{if $n= 2r+1$.}\cr } $$ The concept of Hückel energy was introduced by Coulson as it gives a good approximation for the $\pi$-electron energy of molecular graphs. We obtain two upper bounds and a lower bound for HE$(G)$. When $n$ is even, it is shown that equality holds in both upper bounds if and only if $G$ is a strongly regular graph with parameters $(n, k, \lambda, \mu) = (4t^2 +4t +2,\, 2t^2 +3t +1,\, t^2 +2t,\, t^2 + 2t +1),$ for positive integer $t$. Furthermore, we will give an infinite family of these strongly regular graph whose construction was communicated by Willem Haemers to us. He attributes the construction to J.$\,$J. Seidel.

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 Critical group structure from the parameters of a strongly regular graph

2021 ◽  
Vol 180 ◽  
pp. 105424
Author(s):  
Joshua E. Ducey ◽  
David L. Duncan ◽  
Wesley J. Engelbrecht ◽  
Jawahar V. Madan ◽  
Eric Piato ◽  
...  
Keyword(s):  
Regular Graph ◽  
Group Structure ◽  
Strongly Regular Graph ◽  
Critical Group ◽  
Strongly Regular
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