### Topological Graphs Based on a new Topology on Z^n and its Applications

#### Abstract

Up to now there is no homotopy for Marcus-Wyse (for short $M$-) topological spaces.

In relation to the development of a homotopy for the category of Marcus-Wyse (for short $M$-) topological spaces on ${\bf Z}^2$,

we need to generalize the $M$-topology on ${\bf Z}^2$ to higher dimensional spaces $X \subset {\bf Z}^n$, $n \geq 3$ \cite{HL1}.

Hence the present paper establishes a new topology on ${\bf Z}^n, n \in {\bf N}$, where ${\bf N}$ is the set of natural numbers.

It is called the {\it generalized Marcus-Wyse} (for short $H$-) topology and is denoted by $({\bf Z}^n, \gamma^n)$.

Besides, we prove that $({\bf Z}^3, \gamma^3)$ induces only $6$- or $18$-adjacency relations. Namely, $({\bf Z}^3, \gamma^3)$ does not support a $26$-adjacency, which is quite different from the Khalimsky topology for $3$D digital spaces.

After developing an $H$-adjacency induced by the connectedness of $({\bf Z}^n, \gamma^n)$,

the present paper establishes topological graphs based on the $H$-topology, which is called an $HA$-space in the paper, so that we can establish a category of $HA$-spaces.

By using the $H$-adjacency, we propose an $H$-topological graph homomorphism (for short $HA$-map) and an $HA$-isomorphism.

Besides, we prove that an $HA$-map ({\it resp.} an $HA$-isomorphism) is broader than an $H$-continuous map ({\it resp.} an $H$-homeomorphism) and is an $H$-connectedness preserving map. Finally, after investigating some properties of an $HA$-isomorphism,

we propose both an $HA$-retract and an extension problem of an $HA$-map for studying $HA$-spaces.

In relation to the development of a homotopy for the category of Marcus-Wyse (for short $M$-) topological spaces on ${\bf Z}^2$,

we need to generalize the $M$-topology on ${\bf Z}^2$ to higher dimensional spaces $X \subset {\bf Z}^n$, $n \geq 3$ \cite{HL1}.

Hence the present paper establishes a new topology on ${\bf Z}^n, n \in {\bf N}$, where ${\bf N}$ is the set of natural numbers.

It is called the {\it generalized Marcus-Wyse} (for short $H$-) topology and is denoted by $({\bf Z}^n, \gamma^n)$.

Besides, we prove that $({\bf Z}^3, \gamma^3)$ induces only $6$- or $18$-adjacency relations. Namely, $({\bf Z}^3, \gamma^3)$ does not support a $26$-adjacency, which is quite different from the Khalimsky topology for $3$D digital spaces.

After developing an $H$-adjacency induced by the connectedness of $({\bf Z}^n, \gamma^n)$,

the present paper establishes topological graphs based on the $H$-topology, which is called an $HA$-space in the paper, so that we can establish a category of $HA$-spaces.

By using the $H$-adjacency, we propose an $H$-topological graph homomorphism (for short $HA$-map) and an $HA$-isomorphism.

Besides, we prove that an $HA$-map ({\it resp.} an $HA$-isomorphism) is broader than an $H$-continuous map ({\it resp.} an $H$-homeomorphism) and is an $H$-connectedness preserving map. Finally, after investigating some properties of an $HA$-isomorphism,

we propose both an $HA$-retract and an extension problem of an $HA$-map for studying $HA$-spaces.

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