In this paper we generalize the absolute bound for association schemes to coherent configurations. We examine this bound in the context of quasi-symmetric and strongly regular designs. In particular, we use it to derive a new feasibility condition for strongly regular designs and give examples of parameter sets ruled out by this condition but which pass all other feasibility conditions known to...

We analyze when the Moore–Penrose inverse of the combinatorial Laplacian of a distance– regular graph is a M–matrix; that is, it has non–positive off–diagonal elements. In particular, our results include some previously known results on strongly regular graphs.

An important property of strongly regular graphs is that the second subconstituent of any primitive strongly regular graph is always connected. Brouwer asked to what extent this statement can be generalized to distanceregular graphs. In this paper, we show that if γ is any vertex of a distanceregular graph Γ and t is the index where the standard sequence corresponding to the second largest eige...

Antipodal covers of strongly regular graphs which are not necessarily distance-regular are studied. The structure of short cycles in an antipodal cover is considered. In most cases, this provides a tool to determine if a strongly regular graph has an antipodal cover. In these cases, covers cannot be distance-regular except when they cover a complete bipartite graph. A relationship between antip...

Let be an antipodal distance-regular graph of diameter 4, with eigenvalues θ0 > θ1 > θ2 > θ3 > θ4. Then its Krein parameter q4 11 vanishes precisely when is tight in the sense of Jurišić, Koolen and Terwilliger, and furthermore, precisely when is locally strongly regular with nontrivial eigenvalues p := θ2 and −q := θ3. When this is the case, the intersection parameters of can be parametrized b...

We consider a distance-regular graph ? with diameter d 3 and eigenvalues k = 0 > 1 > > d. We show the intersection numbers a 1 ; b 1 satisfy (a 1 + 1) 2 : We say ? is tight whenever ? is not bipartite, and equality holds above. We characterize the tight property in a number of ways. For example, we show ? is tight if and only if the intersection numbers are given by certain rational expressions...

The inequality of Higman for generalized quadrangles of order (s, t) with s > 1 states that t ≤ s. We will generalize this by proving that the intersection number ci of a regular near 2d-gon of order (s, t) with s > 1 satisfies the tight bound ci ≤ (s − 1)/(s − 1), and we give properties in case of equality. It is known that hemisystems in generalized quadrangles meeting the Higman bound induce...

We consider continuous-time quantum walks on distance-regular graphs. Using results about the existence of complex Hadamard matrices in association schemes, we determine which of these graphs have quantum walks that admit uniform mixing. First we apply a result due to Chan to show that the only strongly regular graphs that admit instantaneous uniform mixing are the Paley graph of order nine and...

A spin model is a square matrix that encodes the basic data for a statistical mechanical construction of link invariants due to V.F.R. Jones. Every spin model W is contained in a canonical Bose-Mesner algebra N (W ). In this paper we study the distance-regular graphs whose Bose-Mesner algebra M satisfies W ∈ M ⊆ N (W ). Suppose W has at least three distinct entries. We show that is 1-homogeneou...

In this paper, we are giving MATLAB program to find the energy and Wiener index of unit graphs. Our program demonstrates an intrinsic relationship between the elements of ring and structural properties of graphs. The unit graph G(Zn) turns out to be strongly regular when n = 2k. Several new directions for further research are also indicated by means of raising problems.