Closed category

The Paley graph of order 13, a strongly regular graph with parameters srg(13,6,2,3).

Bonus: WP Twin and WP Twin Auto Backup: (link to ) While not a theme, I think this software is essential if you are maintaining your Wordpress blog or regularly create new blog sites. You can either install Word - Press yourself or use free services offered on the web today. A pinch of tablet centric strategy can get your Word - Press site miles ahead of your competitors, so here are few strategies that will give your Wordpress websites and blogs an edge over your competitors:. Donor oocytes and menopausal pregnancy: Oocyte donation to women of advanced reproductive age: pregnancy results and obstetrical outcomes in patients 45 years and older. If you are happy with your new look then click "Activate 'New Theme'" in the top right corner.

Word - Press is known as the most popular blogging platform all over the web and is used by millions of blog enthusiasts worldwide. If you are a positive thinker businessman then today you have to put your business online. Our Daily Deal Software plugin brings the simplicity of setting up a Word - Press blog to the daily deal space. t need to use the back button or the URL to get to your home page. Word - Press makes it possible to successfully and manage your website.

Minor and medium sized corporations also have a lot to obtain by shelling out in a very good website. Note: at a first glance WP Mobile Pro themes do not appear to be glamorous or fancy. I've applied numerous Search engine optimization-ready Word - Press themes and I can say from knowledge that I consider the Genesis Search engine marketing panel one particular of the simplest to use. In crux the developer must have a detailed knowledge not only about the marketing tool but also about the ways in which it can be applied profitably. If you've hosted your Word - Press website on a shared hosting server then it'll be easier for you to confirm the restricted access to your site files.

The disadvantage is it requires a considerable amount of time to set every thing up. This plugin allows a webmaster to create complex layouts without having to waste so much time with short codes. A higher percentage of women are marrying at older ages,many are delaying childbearing until their careers are established, the divorce rate is high and many couples remarry and desire their own children. The most important plugins you will need are All-in-One SEO Pack, some social bookmarking plugin, a Feedburner plugin and an RSS sign up button. Look for experience: When you are searching for a Word - Press developer you should always look at their experience level.

There is no denying that Magento is an ideal platform for building ecommerce websites, as it comes with an astounding number of options that can help your online business do extremely well. Being a Plugin Developer, it is important for you to know that development of Word - Press driven website should be done only when you enable debugging. However, you must also manually approve or reject comments so that your website does not promote parasitic behavior. ) Remote Login: With the process of PSD to Wordpress conversion comes the advantage of flexibility. For your information, it is an open source web content management system If you loved this short article and you would certainly like to get even more facts regarding wordpress backup plugin kindly go to our webpage. .

In graph theory, a strongly regular graph is defined as follows. Let G = (V,E) be a regular graph with v vertices and degree k. G is said to be strongly regular if there are also integers λ and μ such that:

• Every two adjacent vertices have λ common neighbours.

• Every two non-adjacent vertices have μ common neighbours.

A graph of this kind is sometimes said to be an srg(v, k, λ, μ).

Some authors exclude graphs which satisfy the definition trivially, namely those graphs which are the disjoint union of one or more equal-sized complete graphs, and their complements, the Turán graphs.

The complement of an srg(v, k, λ, μ) is also strongly regular. It is an srg(v, v−k−1, v−2−2k+μ, v−2k+λ).

A strongly regular graph is a distance-regular graph with diameter 2, but only if μ is non-zero.

Properties

Relationship between Parameters

The four parameters in an srg(v, k, λ, μ) are not independent and must obey the following relation:

$(v-k-1)\mu =k(k-\lambda -1)$

The above relation can be derived very easily through a counting argument as follows:

• Imagine the nodes of the graph to lie in three levels. Pick any node as the root node, in Level 0. Then its k neighbor nodes lie in Level 1, and all other nodes lie in Level 2.
• Nodes in Level 1 are directly connected to the root, hence they must have λ other neighbors in common with the root, and these common neighbors must also be in Level 1. Since each node has degree k, there are $k-\lambda -1$ edges remaining for each Level 1 node to connect to nodes in Level 2.
• Nodes in Level 2 are not directly connected to the root, hence they must have μ common neighbors with the root, and these common neighbors must all be in Level 1. Therefore μ nodes in Level 1 are connected to each node in Level 2 and each of the k nodes in Level 1 is connected to $k-\lambda -1$ nodes in Level 2. Therefore the number of nodes in Level 2 is

${\frac {k(k-\lambda -1)}{\mu }}.$

The total number of nodes across all three levels is therefore

$v=1+k+\left({\frac {k(k-\lambda -1)}{\mu }}\right),$

and by rearranging we get:

$(v-k-1)\mu =k(k-\lambda -1).$

Adjacency Matrix

Let I denote the identity matrix (of order v) and let J denote the matrix whose entries all equal 1. The adjacency matrix A of a strongly regular graph satisfies these properties :

$AJ=kJ$

This is a trivial restatement of the vertex degree requirement.

${A}^{2}+(\mu -\lambda ){A}+(\mu -k){I}=\mu {J}$

The first term gives the number of 2-step paths from each vertex to all vertices. For the vertex pairs directly connected by an edge, the equation reduces to the number of such 2-step paths being equal to λ. For the vertex pairs not directly connected by an edge, the equation reduces to the number of such 2-step paths being equal to μ. For the trivial self-pairs, the equation reduces to the degree being equal to k.

Eigenvalues

• The graph has exactly three eigenvalues:
  • k whose multiplicity is 1
  • ${\frac {1}{2}}\left[(\lambda -\mu )+{\sqrt {(\lambda -\mu )^{2}+4(k-\mu )}}\right]$ whose multiplicity is ${\frac {1}{2}}\left[(v-1)-{\frac {2k+(v-1)(\lambda -\mu )}{\sqrt {(\lambda -\mu )^{2}+4(k-\mu )}}}\right]$
  • ${\frac {1}{2}}\left[(\lambda -\mu )-{\sqrt {(\lambda -\mu )^{2}+4(k-\mu )}}\right]$ whose multiplicity is ${\frac {1}{2}}\left[(v-1)+{\frac {2k+(v-1)(\lambda -\mu )}{\sqrt {(\lambda -\mu )^{2}+4(k-\mu )}}}\right]$

• Strongly regular graphs for which $2k+(v-1)(\lambda -\mu )=0$ are called conference graphs because of their connection with symmetric conference matrices. Their parameters reduce to

${\text{srg}}\left(v,{\tfrac {1}{2}}(v-1),{\tfrac {1}{4}}(v-5),{\tfrac {1}{4}}(v-1)\right).$

• Strongly regular graphs for which $2k+(v-1)(\lambda -\mu )\neq 0$ have integer eigenvalues with unequal multiplicities.

Examples

• The cycle of length 5 is an srg(5,2,0,1).
• The Petersen graph is an srg(10,3,0,1).
• The Clebsch graph is an srg(16,5,0,2).
• The Shrikhande graph is an srg(16,6,2,2) which is not a distance-transitive graph.
• The Line graph of generalized quadrangle GQ(2,4) are srg(27,10,1,5).
• The Schläfli graph is an srg(27,16,10,8).
• The Chang graphs are srg(28,12,6,4).
• The Hoffman–Singleton graph is an srg(50,7,0,1).
• The Sims-Gewirtz graph is an (56,10,0,2).
• The M22 graph is an srg(77,16,0,4).
• The Brouwer–Haemers graph is an srg(81,20,1,6).
• The Higman–Sims graph is an srg(100,22,0,6).
• The Local McLaughlin graph is an srg(162,56,10,24).
• The Paley graph of order q is an srg(q, (q − 1)/2, (q − 5)/4, (q − 1)/4).
• The n × n square rook's graph is an srg(n², 2n − 2, n − 2, 2).

A strongly regular graph is called primitive if both the graph and its complement are connected. All the above graphs are primitive, as otherwise μ=0 or μ=k.

Moore graphs

The strongly regular graphs with λ=0 are triangle free. Apart from the complete graphs on less than 3 vertices and all complete bipartite graphs the seven listed above are the only known ones. Strongly regular graphs with λ=0 and μ=1 are Moore graphs with girth 5. Again the three graphs given above, with parameters (5,2,0,1), (10,3,0,1) and (50,7,0,1), are the only known ones. The only other possible set of parameters yielding a Moore graph is (3250,57,0,1); it is unknown if such a graph exists, and if so, whether or not it is unique.

See also

• Seidel adjacency matrix

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

• A.E. Brouwer, A.M. Cohen, and A. Neumaier (1989), Distance Regular Graphs. Berlin, New York: Springer-Verlag. ISBN 3-540-50619-5, ISBN 0-387-50619-5
• Chris Godsil and Gordon Royle (2004), Algebraic Graph Theory. New York: Springer-Verlag. ISBN 0-387-95241-1

External links

• Eric W. Weisstein, Mathworld article with numerous examples.
• Gordon Royle, List of larger graphs and families.
• Andries E. Brouwer, Parameters of Strongly Regular Graphs.
• Brendan McKay, Some collections of graphs.
• Ted Spence, Strongly regular graphs on at most 64 vertices.