AUTHOR=Wang Guangfu , Li Chenyang , Wang Fengling 

TITLE=l1-Embeddability Under Gate-Sum Operation of Two l1-Graphs

JOURNAL=Frontiers in Physics

VOLUME=8

YEAR=2020

URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2020.00146

DOI=10.3389/fphy.2020.00146

ISSN=2296-424X

ABSTRACT=<p>An <italic>l</italic><sub>1</sub>-graph is one in which the vertices can be labeled by binary vectors such that the Hamming distance between two binary addresses is, to scale, the distance in the graph between the corresponding vertices. This study was designed to determine whether the gate-sum operation can inherit the <italic>l</italic><sub>1</sub>-embeddability. The subgraph <italic>H</italic> of a graph <italic>G</italic> is called a gate subgraph if, for every vertex <italic>v</italic> ∈ <italic>V</italic>(<italic>G</italic>), there exists a vertex <italic>x</italic> ∈ <italic>V</italic>(<italic>H</italic>) such that for every vertex <italic>u</italic> of <italic>H</italic>, <italic>x</italic> lies on a shortest path from <italic>v</italic> to <italic>u</italic>. The graph <italic>G</italic> is defined as the <italic>gate-sum</italic> of two graphs <italic>G</italic><sub>1</sub> and <italic>G</italic><sub>2</sub> with respect to <italic>H</italic> if <italic>H</italic> is a gate subgraph of at least one of <italic>G</italic><sub>1</sub> and <italic>G</italic><sub>2</sub>, such that <italic>G</italic><sub>1</sub>∪<italic>G</italic><sub>2</sub> = <italic>G</italic>, <italic>G</italic><sub>1</sub>∩<italic>G</italic><sub>2</sub> = <italic>H</italic>, and both <italic>G</italic><sub>1</sub> and <italic>G</italic><sub>2</sub> are isometric subgraphs of <italic>G</italic>. In this article, we have shown that the gate-sum graph of two <italic>l</italic><sub>1</sub>-graphs is also an <italic>l</italic><sub>1</sub>-graph.</p>