Reflexivity in some classes of multicyclic treelike graphs
Download
Report
Transcript Reflexivity in some classes of multicyclic treelike graphs
Reflexivity in some classes of
multicyclic treelike graphs
Bojana Mihailović, Zoran Radosavljević, Marija Rašajski
Faculty of Electrical Engineering,
University of Belgrade, Serbia
Introduction
Graph = simple graph (finite, nonoriented, without loops
and/or multiple edges) + connected graph
Spectrum = spectrum of (0,1) adjacency matrix (the
spectrum of a disconnected graph is the union of the
spectra of its components)
A graph is treelike or cactus if any pair of its cycles has
at most one common vertex
A graph is reflexive if its second largest eigenvalue does
not exceed 2
Introduction
Being reflexive is a hereditary property
Presentation of all reflexive graphs inside given set:
via maximal graphs or via minimal forbidden graphs
Smith graphs
n
n-1
1
2
1
2
3
Cn
Wn
n
Instruments
Interlacing theorem
Let A be a symmetric matrix with eigenvalues
1 ,..., n and B one of its principal submatrices
with eigenvalues 1 ,..., m . Then the inequalities
nmi i i (i 1,..., m) hold.
Schwenk’s formulae
newGRAPH
Instruments
RS theorem
Let G be a graph with a cut-vertex u.
1) If at least two components of G-u
are supergraphs of Smith graphs, and
if at least one of them is a proper
supergraph, then 2 (G) 2.
2) If at least two components of G-u
G1
are Smith graphs and the rest are
subgraphs of Smith graphs, then 2 (G) 2.
3) If at most one component of G-u is
a Smith graph, and the rest are proper
subgraphs of Smith graphs, then 2 (G) 2.
G
u
G2
G3
Gn
First results
Class of bicyclic graphs with a bridge between two
cycles of arbitrary length
Additionally loaded vertices which belongs to the
bridge – 36 maximal graphs
Also additionally loaded other vertices – 66 maximal
graphs
First results
Splitting
If we form a tree T by identifying vertices x and y ( x = y = u )
of two trees T1 and T2 , respectively, we may say that the tree T
can be split at its vertex u into T1 and T2 .
x
T1
u
y
T2
T1
T2
First results
Pouring
If we split a tree T at all its vertices u, in all possible ways, and in
each case attach the parts at splitting vertices x and y to
some vertices u and v of a graph G (i.e. identify x with u and
y with v), we say that in the obtained family of graphs the tree T is
pouring between the vertices u and v (including attaching of the
intact tree T, at each vertex, to u or v).
T1
u
G
T2
v
First results
S1
S2
Multicyclic treelike reflexive
graphs
Under 2 conditions:
cut vertex theorem can not be applied
cycles do not form a bundle
treelike reflexive graph has at most 5 cycles.
Multicyclic treelike reflexive
graphs
Under previous 2 conditions
all maximal reflexive cacti with four cycles are determined
four characteristic classes of tricyclic reflexive graphs are
defined
class K 4 is completely described via maximal graphs
l4
K1
K2
K3
K4
New results/current
investigations
classes K1 and K3 are completely described
some new interrelations between these classes and
certain classes of bicyclic and unicyclic graphs are
established
some results are generalized
New results/bundle
cut-vertex theorem can not
be applied, but cycles do
form a bundle
after removing vertex v one
of the components is a
supergraph and all others
subgraphs of some Smith
tree
If G is reflexive, what is
the maximal number of
cycles in it?
G
C2
v
C1
T1
T2
Cn
Tm
New results/bundle
K = the component of the graph G-v which is a supergraph
of some Smith tree
K = minimal component e.g. for every its vertex x, whose
degree in the graph G is 1, condition
holds
2 cases:
K is a subgraph of the cycle C (C is additionally loaded with
1.
some new edges)
K is a subgraph of the tree T
2.
1 ( K x) 2 1 ( K ) (1)
K must contain one of the F - trees (minimal forbidden trees
for
2 2 )
New results/bundle
x
F1
F3
F2
F7
x
F5
F4
x
x
F6
x
x
x
x
F8
x
F9
New results/bundle
1. case
Black vertices are the vertices of K adjacent to vertex v.
both black vertices belong to the same F-tree
K=F
i)
ii)
one black vertex belong to F-tree, and the other doesn’t
any vertex of F-tree different from x may be black
vertex
K = Fi extended with additional path at vertex x
i
4
8
7
3
2
9
path
length
1
1
1,2
1,2,3
arb.
arb.
New results/bundle
2. case
It is sufficient to discuss the case when T-v has one component K.
Black vertex d is a vertex of K adjacent to v.
d belongs to F-tree
i)
ii)
any vertex of F-tree different from x may be black
vertex
K=F
Both cases
PG (2) 0 2 (G) 2
New results/bundle
1. case
C – cycle which contains K; v – cut vertex; x,y – black vertices
PG (2) 2 PC v (2)n1...nk ( PC v x (2) PC v y (2))n1...nk
2 PC v (2)((n1 1)n2 ...nk n1 (n2 1)...nk ... n1...nk 1 ( nk 1))
2 PC Cm (2)n1...nk 2 PC v (2)(n2 ...nk n1n3 ...nk ... n1...nk 1 )
n1...nk (2 PC v (2) PC v x (2) PC v y (2) 2kPC v (2) 2 PC Cm (2))
PG (2) 0 PC (2) 2kPC v (2) 0
New results/bundle
2. case
T-v=K; v – cut vertex
PG (2) 2n1...nk PK (2) 2 PK (2)((n1 1)n2 ...nk
n1 (n2 1)...nk ... n1...nk 1 (nk 1)) PK d (2)n1...nk
2 PK (2)(n2 ...nk n1n3 ...nk ... n1...nk 1 )
n1...nk (2(1 k ) PK (2) PK d (2))
PG (2) 0 2(1 k ) PK (2) PK d (2) 0
New results/bundle
1. case
F1
F2
F3
F4
F5
F6
F7
F8
F9
4
10
12
13
13
13
20
34
74
2. case
F1
F2
F3
F4
F5
F6
F7
F8
F9
2
4
4
4
4
4
7
11
22
New results/bundle
1. case
Maximal number of cycles is 74.
2. case
Maximal number of cycles is 22.
References
D. Cvetković, L. Kraus, S. Simić: Discussing graph theory with a
computer, Implementation of algorithms. Univ. Beograd, Publ.
Elektrotehn. Fak., Ser. Mat. Fiz. No 716 - No 734 (1981), 100104.
B. Mihailović, Z. Radosavljević: On a class of tricyclic reflexive
cactuses. Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. 16
(2005), 55-63.
M. Petrović, Z. Radosavljević: Spectrally constrained graphs.
Fac. of Science, Kragujevac, Serbia, 2001.
Z. Radosavljević, B. Mihailović, M. Rašajski: Decomposition of
Smith graphs in maximal reflexive cacti, Discrete Math., Vol. 308
(2008), 355-366.
Z. Radosavljević, B. Mihailović, M. Rašajski: On bicyclic reflexive
graphs, Discrete Math., Vol. 308 (2008), 715-725.