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The Dynamics of the Forest Graph Operator | | Suresh Dara
; S. M. Hegde
; Venkateshwarlu Deva
; S. B. Rao
; Thomas Zaslavsky
; | | Date: |
6 Jan 2016 | | Abstract: | In 1966, Cummins introduced the "tree graph": the tree graph $mathbf{T}(G)$
of a graph $G$ (possibly infinite) has all its spanning trees as vertices, and
distinct such trees correspond to adjacent vertices if they differ in just one
edge, i.e., two spanning trees $T_1$ and $T_2$ are adjacent if $T_2 = T_1 -e
+f$ for some edges $ein T_1$ and $f
otin T_1$. The tree graph of a connected
graph need not be connected. To obviate this difficulty we define the "forest
graph": let $G$ be a labeled graph of order $alpha$, finite or infinite, and
let $mathfrak{N}(G)$ be the set of all labeled maximal forests of $G$. The
forest graph of $G$, denoted by $mathbf{F}(G)$, is the graph with vertex set
$mathfrak{N}(G)$ in which two maximal forests $F_1$, $F_2$ of $G$ form an edge
if and only if they differ exactly by one edge, i.e., $F_2 = F_1 -e +f$ for
some edges $ein F_1$ and $f
otin F_1$.
Using the theory of cardinal numbers, Zorn’s lemma, transfinite induction,
the axiom of choice and the well-ordering principle, we determine the
$mathbf{F}$-convergence, $mathbf{F}$-divergence, $mathbf{F}$-depth and
$mathbf{F}$-stability of any graph $G$. In particular it is shown that a graph
$G$ (finite or infinite) is $mathbf{F}$-convergent if and only if $G$ has at
most one cycle of length 3. The $mathbf{F}$-stable graphs are precisely $K_3$
and $K_1$. The $mathbf{F}$-depth of any graph $G$ different from $K_3$ and
$K_1$ is finite. We also determine various parameters of $mathbf{F}(G)$ for an
infinite graph $G$, including the number, order, size, and degree of its
components. | | Source: | arXiv, 1601.1041 | | Services: | Forum | Review | PDF | Favorites | |
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