233 research outputs found
The edge-flipping group of a graph
Let be a finite simple connected graph with vertices and
edges. A configuration is an assignment of one of two colors, black or white,
to each edge of A move applied to a configuration is to select a black
edge and change the colors of all adjacent edges of
Given an initial configuration and a final configuration, try to find a
sequence of moves that transforms the initial configuration into the final
configuration. This is the edge-flipping puzzle on and it corresponds to a
group action. This group is called the edge-flipping group of
This paper shows that if has at least three vertices,
is isomorphic to a semidirect product of
and the symmetric group of degree where
if is odd, if is even, and
is the additive group of integers.Comment: 19 page
The flipping puzzle on a graph
Let be a connected graph which contains an induced path of
vertices, where is the order of We consider a puzzle on . A
configuration of the puzzle is simply an -dimensional column vector over
with coordinates of the vector indexed by the vertex set . For
each configuration with a coordinate , there exists a move that
sends to the new configuration which flips the entries of the coordinates
adjacent to in We completely determine if one configuration can move
to another in a sequence of finite steps.Comment: 18 pages, 1 figure and 1 tabl
Distance-regular graphs, pseudo primitive idempotents, and the Terwilliger algebra
Let denote a distance-regular graph with diameter and
Bose-Mesner algebra . For we define a 1 dimensional
subspace of which we call . If then
consists of those in such that , where
(resp. ) is the adjacency matrix (resp. th distance matrix) of
If then . By a {\it pseudo
primitive idempotent} for we mean a nonzero element of . We
use pseudo primitive idempotents to describe the irreducible modules for the
Terwilliger algebra, that are thin with endpoint one.Comment: 17 page
3-bounded property in a triangle-free distance-regular graph
Let denote a distance-regular graph with classical parameters and . Assume the intersection numbers and
. We show is 3-bounded in the sense of the article
[D-bounded distance-regular graphs, European Journal of Combinatorics(1997)18,
211-229].Comment: 13 page
Strongly Regular Graphs as Laplacian Extremal Graphs
The Laplacian spread of a graph is the difference between the largest
eigenvalue and the second-smallest eigenvalue of the Laplacian matrix of the
graph. We find that the class of strongly regular graphs attains the maximum of
largest eigenvalues, the minimum of second-smallest eigenvalues of Laplacian
matrices and hence the maximum of Laplacian spreads among all simple connected
graphs of fixed order, minimum degree, maximum degree, minimum size of common
neighbors of two adjacent vertices and minimum size of common neighbors of two
nonadjacent vertices. Some other extremal graphs are also provided.Comment: 11 pages, 4 figures, 1 tabl
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