scholarly journals The Smallest Strictly Neumaier Graph and its Generalisations

10.37236/8189 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Rhys J. Evans ◽  
Sergey Goryainov ◽  
Dmitry Panasenko
Keyword(s):  
Positive Integer ◽  
Regular Graph ◽  
Regular Graphs ◽  
Current Interest ◽  
Polar Graph ◽  
Strongly Regular ◽  
Regular Cliques

A regular clique in a regular graph is a clique such that every vertex outside of the clique is adjacent to the same positive number of vertices inside the clique. We continue the study of regular cliques in edge-regular graphs initiated by A. Neumaier in the 1980s and attracting current interest. We thus define a Neumaier graph to be an non-complete edge-regular graph containing a regular clique, and a strictly Neumaier graph to be a non-strongly regular Neumaier graph. We first prove some general results on Neumaier graphs and their feasible parameter tuples. We then apply these results to determine the smallest strictly Neumaier graph, which has $16$ vertices. Next we find the parameter tuples for all strictly Neumaier graphs having at most $24$ vertices. Finally, we give two sequences of graphs, each with $i^{th}$ element a strictly Neumaier graph containing a $2^{i}$-regular clique (where $i$ is a positive integer) and having parameters of an affine polar graph as an edge-regular graph. This answers questions recently posed by G. Greaves and J. Koolen.

scholarly journals Extension of Strongly Regular Graphs

10.37236/878 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Ralucca Gera ◽  
Jian Shen
Keyword(s):  
Positive Integer ◽  
Nonnegative Integer ◽  
Regularity Property ◽  
Strongly Regular Graphs ◽  
Regular Graphs ◽  
Strongly Regular ◽  
Vertex Degrees

The Friendship Theorem states that if any two people in a party have exactly one common friend, then there exists a politician who is a friend of everybody. In this paper, we generalize the Friendship Theorem. Let $\lambda$ be any nonnegative integer and $\mu$ be any positive integer. Suppose each pair of friends have exactly $\lambda$ common friends and each pair of strangers have exactly $\mu$ common friends in a party. The corresponding graph is a generalization of strongly regular graphs obtained by relaxing the regularity property on vertex degrees. We prove that either everyone has exactly the same number of friends or there exists a politician who is a friend of everybody. As an immediate consequence, this implies a recent conjecture by Limaye et. al.

scholarly journals On the Resistance Matrix of a Graph

10.37236/5295 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Jiang Zhou ◽  
Zhongyu Wang ◽  
Changjiang Bu
Keyword(s):  
Regular Graph ◽  
Connected Graph ◽  
Spanning Trees ◽  
Regular Graphs ◽  
Resistance Distance ◽  
Strongly Regular ◽  
Resistance Matrix ◽  
Number Of Spanning Trees

Let $G$ be a connected graph of order $n$. The resistance matrix of $G$ is defined as $R_G=(r_{ij}(G))_{n\times n}$, where $r_{ij}(G)$ is the resistance distance between two vertices $i$ and $j$ in $G$. Eigenvalues of $R_G$ are called R-eigenvalues of $G$. If all row sums of $R_G$ are equal, then $G$ is called resistance-regular. For any connected graph $G$, we show that $R_G$ determines the structure of $G$ up to isomorphism. Moreover, the structure of $G$ or the number of spanning trees of $G$ is determined by partial entries of $R_G$ under certain conditions. We give some characterizations of resistance-regular graphs and graphs with few distinct R-eigenvalues. For a connected regular graph $G$ with diameter at least $2$, we show that $G$ is strongly regular if and only if there exist $c_1,c_2$ such that $r_{ij}(G)=c_1$ for any adjacent vertices $i,j\in V(G)$, and $r_{ij}(G)=c_2$ for any non-adjacent vertices $i,j\in V(G)$.

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 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 Irregularity Strength of Regular Graphs

10.37236/806 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Jakub Przybyło
Keyword(s):  
Positive Integer ◽  
Recent Result ◽  
Regular Graph ◽  
Simple Graph ◽  
Regular Graphs ◽  
Isolated Vertex ◽  
Irregularity Strength

Let $G$ be a simple graph with no isolated edges and at most one isolated vertex. For a positive integer $w$, a $w$-weighting of $G$ is a map $f:E(G)\rightarrow \{1,2,\ldots,w\}$. An irregularity strength of $G$, $s(G)$, is the smallest $w$ such that there is a $w$-weighting of $G$ for which $\sum_{e:u\in e}f(e)\neq\sum_{e:v\in e}f(e)$ for all pairs of different vertices $u,v\in V(G)$. A conjecture by Faudree and Lehel says that there is a constant $c$ such that $s(G)\le{n\over d}+c$ for each $d$-regular graph $G$, $d\ge 2$. We show that $s(G) < 16{n\over d}+6$. Consequently, we improve the results by Frieze, Gould, Karoński and Pfender (in some cases by a $\log n$ factor) in this area, as well as the recent result by Cuckler and Lazebnik.

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 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 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.

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. .

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.

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