What graph parameters are determined by parameters for strongly regular graph - MathOverflow

What graph parameters are determined by parameters for strongly regular graph

2012-12-13T01:17:47Z

Say two graphs are not isomorphic but are both strongly regular with the same set of parameters. Are there any parameters (other than the usual such as order, degrees, eigenvalues and multiplicities, etc.) that are determined, e.g., independence number, chromatic number, etc.?

Thanks for any help

Answer by Graphth for What graph parameters are determined by parameters for strongly regular graph

2012-12-13T01:46:58Z

Okay, well, I checked Brouwer's website and combined that with the comment to the accepted answer of a question on this site. I checked the complement of the Shrikhande graph versus the complement of the line graph of $K_{4, 4}$ using Sage and found independence numbers of 3 and 4, and chromatic numbers of 6 and 4, respectively. Both are strongly regular with parameters (16, 9, 4, 6). So, that answers my question for some parameters.

They have the same girth though.

Answer by Dima Pasechnik for What graph parameters are determined by parameters for strongly regular graph

2012-12-13T09:41:41Z

It's a classic result that a graph parameter called Lovasz theta-function $\theta(\Gamma)$ of a strongly regular graph $\Gamma$ is determined by its parameters. And the significance of $\theta(\Gamma)$ is that it is "sandwiched" between the clique number and the chromatic number.

In more detail, the parameters of the s.r.g. $\Gamma$ determine a 3-dimensional commutative algebra of symmetric matrices (the adjacency matrix $A(\Gamma)$ of $\Gamma$, the adjacency matrix of its complement, and the identity matrix span this algebra). Anything that can be expressed in terms of this algebra, which is specified by the eigenvalues of $A(\Gamma)$, is a parameter you are asking about, and $\theta(\Gamma)$ is one of them. Another one is the number of spanning trees, as by Matrix Tree Theorem it is determined by the eigenvalues.

Answer by Louis Deaett for What graph parameters are determined by parameters for strongly regular graph

2012-12-20T15:25:25Z

It seems the girth of a strongly regular graph would be determined by its parameters in the following way. If $\lambda > 0$, then the girth is 3. If $\lambda=0$ and $\mu > 1$, then the girth is 4. If $\lambda=0$ and $\mu=1$ then the girth is 5. That last case is a little unusual...

Answer by Mohsen for What graph parameters are determined by parameters for strongly regular graph

2012-12-21T10:28:12Z

The diameter, energy and number of closed walks could be determined by parameters.

Answer by Brendan McKay for What graph parameters are determined by parameters for strongly regular graph

2012-12-21T11:39:46Z

The number of cycles of length 3,4,5 are determined. If the girth is 4, the number of 6-cycles is determined too.