Let Γ be a distance-regular graph of diameter d , and t be an integer with 2 ≤ t ≤ d − 1 such that at−1 = 0. For any pair (u, v) of vertices, let Π (u, v) be the subgraph induced by the vertices lying on shortest paths between u and v. We prove that if Π (u, v) is a bipartite geodetically closed subgraph for some pair (u, v) of vertices at distance t , then Π (x, y) is a bipartite geodetically ...

Girth pairs were introduced by Harary and Kovács [Regular graphs with given girth pair, J. Graph Theory 7 (1983) 209–218]. The odd girth (even girth) of a graph is the length of a shortest odd (even) cycle. Let g denote the smaller of the odd and even girths, and let h denote the larger. Then (g, h) is called the girth pair of the graph. In this paper we prove that a graph with girth pair (g, h...

In a recent article [Gi99], V.Ginzburg introduced and studied in depth the notion of a principal nilpotent pair in a semisimple Lie algebra g. He also obtained several results for more general pairs. As a next step, we considered in [Pa99] almost principal nilpotent pairs. The aim of this paper is to make a contribution to the general theory of nilpotent pairs. Roughly speaking, a nilpotent pai...

Anyonic states, which are topologically robust because of their peculiar structure Hilbert space, have important applications in quantum computing and communication. Here we investigate the capacity deterministic one-time pad (DQOTP) that uses Fibonacci anyons as an information carrier. We find particle-antiparticle pair produced from vacuum can be used to asymptotically send $2{log}_{2}{d}_{\e...

A groupoid is a small category in which all morphisms are isomorphisms. An inductive specialized whose object set regular biordered and the admit partial order. normal strict preorder factorization property. pair of ‘related’ categories constitutes cross-connection. Both groupoids cross-connections were identified by Nambooripad as categorical models semigroups. We explore inter-relationship be...

Torus with a single 3-5 pair. The nonexistence of a quadrangulation with a single 3-5 pair for a torus has been proved in [Barnette et al. 1971]. Remarkably, [Jucovič and Trenkler 1973] shows that for any choice of non-regular vertex valences, constrained by the formula for Euler’s characteristics, and any genus other than 1, a quadrangulation with vertices with these valences exists (if vertic...

According to the classical Friendship Theorem, if G is a finite simple graph such that each two distinct vertices have a unique common neighbor, then G has a dominating vertex. Jian Shen showed that if require only non-neighbors to have a unique common neighbor, then one of the following possibilities hold: i) G has dominating vertex, or ii) G is a strongly regular graph, or iii) G is a bi-regu...

In this paper we extend the well known Kalman-Yacubovic-Popov (KYP) lemma to the case of matrices with general regular inertia. We show that the version of the lemma that was derived for the case of pairs of stable matrices whose rank difference is one, extends to the more general case of matrices with regular inertia and in companion form. We then use this result to derive an easily verifiable...

A oterie is a family of subsets su h that every pair of subsets has at least one element in ommon but neither is a subset of the other. A oterie C is said to be non-dominated (ND) if there is no other oterie D su h that, for 8Q 2 C, there exists Q0 2 D satisfying Q0 Q. We introdu e an operator , whi h transforms a ND oterie to another ND oterie. A regular oterie is a natural generalization of a...