Hi, I have a question about the relationship between parameters of weakly regular graphs (if it exists)
Consider two weakly regular graphs with parameters (n,k,lambda,beta) where n is number of vertices, k is the degree, lambda is the number of common neighbors of adjacent vertices, and beta is the number of common neighbors of nonadjacent vertices. (At least one of the latter two parameters (lambda,beta) will have two values for a weakly regular graph).
Suppose (n,k) is the same for both graphs. And suppose one graph has lambda=0 for all vertices (zero clustering, triangle-free) and the other one has lambda=1 for all.. Can we say/show that when lambda is 1 (or in general higher), beta's of the remaining vertices in that graph is lower than the other graph? In other words, the number of common neighbours of nonadjacent vertices is less when the number of common neighbours of adjacent vertices are higher in a graph.
thank you very much in advance..