
TL;DR
This paper studies the structure of $oldsymbol{ extit{ ext{ell}}}$-link graphs and their roots, providing bounds and characterizations that extend Whitney's isomorphism theorem to a broader class of graph constructs.
Contribution
It introduces a framework for understanding $oldsymbol{ extit{ ext{ell}}}$-roots, showing they decompose into minimal roots and bounded trees, and generalizes Whitney's theorem to these roots.
Findings
Every $oldsymbol{ extit{ ext{ell}}}$-root decomposes into a minimal root and bounded trees.
Bounds are established for the number, size, and degree of minimal $oldsymbol{ extit{ ext{ell}}}$-roots.
Characterizations are provided for $oldsymbol{ extit{ ext{ell}}}$-roots of specific graphs like cycles.
Abstract
Let be an integer, and be a graph without loops. An -link of is a walk of length in which consecutive edges are different. We identify an -link with its reverse sequence. The -link graph of is defined to have vertices the -links of , such that two vertices of are adjacent if their corresponding -links are the initial and final subsequences of an -link of . A graph is called an -root of a graph if . For example, . And the -link graph of a simple graph is the line graph of that graph. Moreover, let be a finite connected simple graph. Whitney's isomorphism theorem (1932) states if has two connected nonnull simple -roots, then , and the two -roots are isomorphic to…
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Taxonomy
TopicsInterconnection Networks and Systems · Advanced Graph Theory Research · graph theory and CDMA systems
