For any graph G with fixed boundary there exists a layout in the plane, which minimizes the maximum Euclidean distance of any node to its neighbors. This layout balances the length of the graph edges and is therefore called a (length-) balanced layout of G. Furthermore the existence of a unique optimal balanced layout L with the following properties is proved: (i) L is the minimal element of an...

We give a new bound on the parameter λ (number of common neighbors of a pair of adjacent vertices) in a distance-regular graph G, improving and generalizing bounds for strongly regular graphs by Spielman (1996) and Pyber (2014). The new bound is one of the ingredients of recent progress on the complexity of testing isomorphism of strongly regular graphs (Babai, Chen, Sun, Teng, Wilmes 2013). Th...

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...

Delsarte showed that for any projective linear code over a finite field GF (p) with two nonzero Hamming weights w1 < w2 there exist positive integers u and s such that w1 = p u and w2 = p (u + 1). Moreover, he showed that the additive group of such a code has a strongly regular Cayley graph. Here we show that for any proper, regular, projective linear code C over a finite Frobenius ring with tw...

In this paper, we study a distance-regular graph Γ = (X,R) with an intersection number a2 6= 0 having a strongly closed subgraph Y of diameter 2. Let E0, E1, . . . , ED be the primitive idempotents corresponding to the eigenvalues θ0 > θ1 > · · · > θD of Γ. Let V = C be the vector space consisting of column vectors whose rows are labeled by the vertex set X. Let W be the subspace of V consistin...

In this paper, we address the Topology control with Cooperative Communication (TCC) problem in ad hoc wireless network systems. Cooperative communication is a peculiar model introduced recently that allows combining partial messages to decode a full message. The goal of the TCC problem is to gain a strongly-connected topology with minimum total energy consumption. We show TCC problem is NP-comp...

The notion of strongly quotient graph (SQG) was introduced by Adiga et al. (2007). In this paper, we obtain some better results for the distance energy and the distance Estrada index of any connected strongly quotient graph (CSQG) as well as some relations between the distance Estrada index and the distance energy. We also present some bounds for the distance energy and the distance Estrada ind...