اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدStrongly regular graphs with the same parameters as the symplectic graph
73
0
0.0
(
0
)
تأليف
Sho Kubota
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We consider orbit partitions of groups of automorphisms for the symplectic graph and apply Godsil-McKay switching. As a result, we find four families of strongly regular graphs with the same parameters as the symplectic graphs, including the one discovered by Abiad and Haemers. Also, we prove that switched graphs are non-isomorphic to each other by considering the number of common neighbors of three vertices.
قيم البحث
اقرأ أيضاً
A known Kronecker construction of completely regular codes has been investigated taking different alphabets in the component codes. This approach is also connected with lifting constructions of completely regular codes. We obtain several classes of c
ompletely regular codes with different parameters, but identical intersection array. Given a prime power $q$ and any two natural numbers $a,b$, we construct completely transitive codes over different fields with covering radius $rho=min{a,b}$ and identical intersection array, specifically, one code over $F_{q^r}$ for each divisor $r$ of $a$ or $b$. As a corollary, for any prime power $q$, we show that distance regular bilinear forms graphs can be obtained as coset graphs from several completely regular codes with different parameters. Under the same conditions, an explicit construction of an infinite family of $q$-ary uniformly packed codes (in the wide sense) with covering radius $rho$, which are not completely regular, is also given.
The Laplacian spread of a graph is the difference between the largest eigenvalue and the second-smallest eigenvalue of the Laplacian matrix of the graph. We find that the class of strongly regular graphs attains the maximum of largest eigenvalues, th
e minimum of second-smallest eigenvalues of Laplacian matrices and hence the maximum of Laplacian spreads among all simple connected graphs of fixed order, minimum degree, maximum degree, minimum size of common neighbors of two adjacent vertices and minimum size of common neighbors of two nonadjacent vertices. Some other extremal graphs are also provided.
In this paper we show that there does not exist a strongly regular graph with parameters $(1911,270,105,27)$.
In this paper, we simplify the known switching theorem due to Bose and Shrikhande as follows. Let $G=(V,E)$ be a primitive strongly regular graph with parameters $(v,k,lambda,mu)$. Let $S(G,H)$ be the graph from $G$ by switching with respect to a
nonempty $Hsubset V$. Suppose $v=2(k-theta_1)$ where $theta_1$ is the nontrivial positive eigenvalue of the $(0,1)$ adjacency matrix of $G$. This strongly regular graph is associated with a regular two-graph. Then, $S(G,H)$ is a strongly regular graph with the same parameters if and only if the subgraph induced by $H$ is $k-frac{v-h}{2}$ regular. Moreover, $S(G,H)$ is a strognly regualr graph with the other parameters if and only if the subgraph induced by $H$ is $k-mu$ regular and the size of $H$ is $v/2$. We prove these theorems with the view point of the geometrical theory of the finite set on the Euclidean unit sphere.
Davis and Jedwab (1997) established a great construction theory unifying many previously known constructions of difference sets, relative difference sets and divisible difference sets. They introduced the concept of building blocks, which played an i
mportant role in the theory. On the other hand, Polhill (2010) gave a construction of Paley type partial difference sets (conference graphs) based on a special system of building blocks, called a covering extended building set, and proved that there exists a Paley type partial difference set in an abelian group of order $9^iv^4$ for any odd positive integer $v>1$ and any $i=0,1$. His result covers all orders of nonelementary abelian groups in which Paley type partial difference sets exist. In this paper, we give new constructions of strongly regular Cayley graphs on abelian groups by extending the theory of building blocks. The constructions are large generalizations of Polhills construction. In particular, we show that for a positive integer $m$ and elementary abelian groups $G_i$, $i=1,2,ldots,s$, of order $q_i^4$ such that $2m,|,q_i+1$, there exists a decomposition of the complete graph on the abelian group $G=G_1times G_2times cdotstimes G_s$ by strongly regular Cayley graphs with negative Latin square type parameters $(u^2,c(u+1),- u+c^2+3 c,c^2+ c)$, where $u=q_1^2q_2^2cdots q_s^2$ and $c=(u-1)/m$. Such strongly regular decompositions were previously known only when $m=2$ or $G$ is a $p$-group. Moreover, we find one more new infinite family of decompositions of the complete graphs by Latin square type strongly regular Cayley graphs. Thus, we obtain many strongly regular graphs with new parameters.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
