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Transcript of Reflexivity in some classes of multicyclic treelike graphs Bojana Mihailović, Zoran Radosavljević,...
![Page 1: Reflexivity in some classes of multicyclic treelike graphs Bojana Mihailović, Zoran Radosavljević, Marija Rašajski Faculty of Electrical Engineering, University.](https://reader036.fdocuments.us/reader036/viewer/2022062500/5697bf8c1a28abf838c8bb82/html5/thumbnails/1.jpg)
Reflexivity in some classes of multicyclic treelike graphs
Bojana Mihailović, Zoran Radosavljević, Marija Rašajski
Faculty of Electrical Engineering, University of Belgrade, Serbia
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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
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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
1
3
2
n n -1
C n W n
1 2 n
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Instruments
Interlacing theorem Let A be a symmetric matrix with eigenvalues and B one of its principal submatrices with eigenvalues Then the
inequalities hold. Schwenk’s formulae newGRAPH
1,..., n 1,..., .m
( 1,..., )n m i i i i m
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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) If at least two components of G-u are Smith graphs and the rest are subgraphs of Smith graphs, then
3) If at most one component of G-u is a Smith graph, and the rest are proper subgraphs of Smith graphs, then
u
1G2G 3G nG
2 ( ) 2.G
2 ( ) 2.G
2 ( ) 2.G
G
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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
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First results Splitting
If we form a tree T by identifying vertices x and y of two trees and , respectively, we may say that the
tree Tcan be split at its vertex u into and .
x y uT 1 T 2 T 1 T 2
( )x = y = u1T 2T
1T 2T
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First results PouringIf 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 andy 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).
GT 1 T 2u v
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First results
1S
2S
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Multicyclic treelike reflexive graphsUnder 2 conditions: cut vertex theorem can not be applied cycles do not form a bundletreelike reflexive graph has at most 5 cycles.
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Multicyclic treelike reflexive graphsUnder previous 2 conditions all maximal reflexive cacti with four cycles are
determined four characteristic classes of tricyclic reflexive graphs
are defined class is completely described via maximal graphs
4l
4K
4K3K2K1K
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New results/current investigations
classes and are completely described some new interrelations between these
classes and certain classes of bicyclic and unicyclic graphs are established
some results are generalized
1K 3K
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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?
1C
2C
nC
1T
2TmT
v
G
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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:
1. K is a subgraph of the cycle C (C is additionally loaded with some new edges)
2. K is a subgraph of the tree T K must contain one of the F - trees (minimal
forbidden trees for )
1 1( ) 2 ( ) (1)K x K
2 2
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New results/bundle
1F 2F 3F 4F 5F
6F7F 8F 9F
xx x x x
x x x x
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New results/bundle
1. caseBlack vertices are the vertices of K adjacent to vertex v. both black vertices belong to the same F-tree
one black vertex belong to F-tree, and the other doesn’t
i) any vertex of F-tree different from x may be black vertex
ii) extended with additional path at vertex x
iK = F
K = F
i 4 8 7 3 2 9
path length
1 1 1,2 1,2,3
arb. arb.
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New results/bundle
2. caseIt 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) any vertex of F-tree different from x may be black vertex
ii) K=F
Both cases 2(2) 0 ( ) 2GP G
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New results/bundle 1. caseC – cycle which contains K; v – cut vertex; x,y – black
vertices 1 1
1 2 1 2 1 1
1 2 1 3 1 1
1
(2) 2 (2) ... ( (2) (2)) ...
2 (2)(( 1) ... ( 1)... ... ... ( 1))
2 (2) ... 2 (2)( ... ... ... ... )
... (2 (2) (2)m
G C v k C v x C v y k
C v k k k k
C C k C v k k k
k C v C v x C v
P P n n P P n n
P n n n n n n n n n
P n n P n n n n n n n
n n P P P
(2) 2 (2) 2 (2))my C v C CkP P
(2) 0 (2) 2 (2) 0G C C vP P kP
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New results/bundle 2. caseT-v=K; v – cut vertex
1 1 2
1 2 1 1 1
2 1 3 1 1
1
(2) 2 ... (2) 2 (2)(( 1) ...
( 1)... ... ... ( 1)) (2) ...
2 (2)( ... ... ... ... )
... (2(1 ) (2) (2))
G k K K k
k k k K d k
K k k k
k K K d
P n n P P n n n
n n n n n n P n n
P n n n n n n n
n n k P P
(2) 0 2(1 ) (2) (2) 0G K K dP k P P
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1F 2F 3F 4F 5F 6F 7F 8F 9F
4 10 12 13 13 13 20 34 74
New results/bundle 1. case
2. case
1F 2F 3F 4F 5F 6F 7F 8F 9F
2 4 4 4 4 4 7 11 22
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New results/bundle 1. case
Maximal number of cycles is 74.
2. case
Maximal number of cycles is 22.
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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), 100-104.
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.