**Authors:** Luigia Berardi, Mario Gionfriddo, Rosaria Rota

### Abstract

An

*octagon quadrangle* is the graph consisting of an 8-cycle (

*x*_{1},...,

*x*_{8}) with two additional chords: the edges {

*x*_{1},

*x*_{4}} and {

*x*_{5},

*x*_{8}}. An

*octagon quadrangle system* of order

*v* and index λ [

*OQS*] is a pair (

*X*,

*Β*), where X is a finite set of

*v* vertices and

*Β* is a collection of edge disjoint octagon quadrangles (called

*blocks*) which partition the edge set of λK

_{v} defined on X. A 4-

*kite* is the graph having five vertices

*x*_{1},

*x*_{2},

*x*_{3},

*x*_{4},

*y* and consisting of an 4-cycle (

*x*_{1},

*x*_{2},...,

*x*_{4}) and an additional edge {

*x*_{1},

*y*}. A 4-

*kite design* of order

*n* and index

*μ* is a pair

*K*=(

*Y*,

*H*), where Y is a finite set of

*n* vertices and

*H* is a collection of edge disjoint 4-

*kite* which partition the edge set of

*μK*_{n} defined on Y. An

*Octagon Kite System* [

*OKS*] of order

*v* and indices (λ, μ) is an

*OQS*(

*v*) of index λ in which it is possible to divide every block in two 4-

*kites* so that an 4-

*kite design* of order

*v* and index μ is defined.

In this paper we determine the spectrum for

*OKS*(

*v*) nesting 4-kite-designs of

*equi-indices* (2,3).

Luigia Berardi Dipartimento di Ingegneria Elettrica e dell'Informazione,

Universitá di L'Aquila

E-mail:

Mario Gionfriddo

Dipartimento di Matematica e Informatica,

Universitá di Catania

E-mail:

Rosaria Rota

Dipartimento di Matematica, Universitá di RomaTre

E-mail:

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