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Group Edge Irregularity Strength of Graphs Marcin Anholcer Pozna - PowerPoint PPT Presentation

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Outline Introduction Group Edge Irregularity Strength The End Group Edge Irregularity Strength of Graphs Marcin Anholcer Pozna University of Economics June 18, 2015, Koper Marcin Anholcer Group Edge Irregularity Strength of Graphs 1/ 32

Outline Introduction Group Edge Irregularity Strength The End Group Edge Irregularity Strength of Graphs Marcin Anholcer Poznań University of Economics June 18, 2015, Koper Marcin Anholcer Group Edge Irregularity Strength of Graphs 1/ 32
Outline Introduction Group Edge Irregularity Strength The End 1 Introduction Notation Irregularity Strength Edge Irregularity Strength Labelling the Graph with Abelian Groups Group Irregularity Strength 2 Group Edge Irregularity Strength Definition Results 3 The End Open Problems Thank You Marcin Anholcer Group Edge Irregularity Strength of Graphs 2/ 32
Notation Outline Irregularity Strength Introduction Edge Irregularity Strength Group Edge Irregularity Strength Labelling the Graph with Abelian Groups The End Group Irregularity Strength Notation G - simple graph E ( G ) - the edge set of G , m = | E ( G ) | V ( G ) - the vertex set of G , n = | V ( G ) | Maximum degree: ∆( G ) , minimum degree: δ ( G ) G - Abelian group, for convenience: 0, 2 a , − a , a − b . . . Marcin Anholcer Group Edge Irregularity Strength of Graphs 3/ 32
Notation Outline Irregularity Strength Introduction Edge Irregularity Strength Group Edge Irregularity Strength Labelling the Graph with Abelian Groups The End Group Irregularity Strength Notation G - simple graph E ( G ) - the edge set of G , m = | E ( G ) | V ( G ) - the vertex set of G , n = | V ( G ) | Maximum degree: ∆( G ) , minimum degree: δ ( G ) G - Abelian group, for convenience: 0, 2 a , − a , a − b . . . Marcin Anholcer Group Edge Irregularity Strength of Graphs 3/ 32
Notation Outline Irregularity Strength Introduction Edge Irregularity Strength Group Edge Irregularity Strength Labelling the Graph with Abelian Groups The End Group Irregularity Strength Notation G - simple graph E ( G ) - the edge set of G , m = | E ( G ) | V ( G ) - the vertex set of G , n = | V ( G ) | Maximum degree: ∆( G ) , minimum degree: δ ( G ) G - Abelian group, for convenience: 0, 2 a , − a , a − b . . . Marcin Anholcer Group Edge Irregularity Strength of Graphs 3/ 32
Notation Outline Irregularity Strength Introduction Edge Irregularity Strength Group Edge Irregularity Strength Labelling the Graph with Abelian Groups The End Group Irregularity Strength Notation G - simple graph E ( G ) - the edge set of G , m = | E ( G ) | V ( G ) - the vertex set of G , n = | V ( G ) | Maximum degree: ∆( G ) , minimum degree: δ ( G ) G - Abelian group, for convenience: 0, 2 a , − a , a − b . . . Marcin Anholcer Group Edge Irregularity Strength of Graphs 3/ 32
Notation Outline Irregularity Strength Introduction Edge Irregularity Strength Group Edge Irregularity Strength Labelling the Graph with Abelian Groups The End Group Irregularity Strength Notation G - simple graph E ( G ) - the edge set of G , m = | E ( G ) | V ( G ) - the vertex set of G , n = | V ( G ) | Maximum degree: ∆( G ) , minimum degree: δ ( G ) G - Abelian group, for convenience: 0, 2 a , − a , a − b . . . Marcin Anholcer Group Edge Irregularity Strength of Graphs 3/ 32
Notation Outline Irregularity Strength Introduction Edge Irregularity Strength Group Edge Irregularity Strength Labelling the Graph with Abelian Groups The End Group Irregularity Strength s ( G ) : Definition Assign positive integer w ( e ) ≤ s to every edge e ∈ E ( G ) . For every vertex v ∈ V ( G ) the weighted degree is defined as: � wd ( v ) = w ( e ) . e ∋ v w is irregular if for v � = u we have wd ( v ) � = wd ( u ) . Irregularity strength s ( G ) : the lowest s that allows some irregular labeling. Introduced by G. Chartrand, M.S. Jacobson, J. Lehel, O.R. Oellermann, S. Ruiz, F. Saba, 1988. Marcin Anholcer Group Edge Irregularity Strength of Graphs 4/ 32
Notation Outline Irregularity Strength Introduction Edge Irregularity Strength Group Edge Irregularity Strength Labelling the Graph with Abelian Groups The End Group Irregularity Strength s ( G ) : Definition Assign positive integer w ( e ) ≤ s to every edge e ∈ E ( G ) . For every vertex v ∈ V ( G ) the weighted degree is defined as: � wd ( v ) = w ( e ) . e ∋ v w is irregular if for v � = u we have wd ( v ) � = wd ( u ) . Irregularity strength s ( G ) : the lowest s that allows some irregular labeling. Introduced by G. Chartrand, M.S. Jacobson, J. Lehel, O.R. Oellermann, S. Ruiz, F. Saba, 1988. Marcin Anholcer Group Edge Irregularity Strength of Graphs 4/ 32
Notation Outline Irregularity Strength Introduction Edge Irregularity Strength Group Edge Irregularity Strength Labelling the Graph with Abelian Groups The End Group Irregularity Strength s ( G ) : Definition Assign positive integer w ( e ) ≤ s to every edge e ∈ E ( G ) . For every vertex v ∈ V ( G ) the weighted degree is defined as: � wd ( v ) = w ( e ) . e ∋ v w is irregular if for v � = u we have wd ( v ) � = wd ( u ) . Irregularity strength s ( G ) : the lowest s that allows some irregular labeling. Introduced by G. Chartrand, M.S. Jacobson, J. Lehel, O.R. Oellermann, S. Ruiz, F. Saba, 1988. Marcin Anholcer Group Edge Irregularity Strength of Graphs 4/ 32
Notation Outline Irregularity Strength Introduction Edge Irregularity Strength Group Edge Irregularity Strength Labelling the Graph with Abelian Groups The End Group Irregularity Strength s ( G ) : Definition Assign positive integer w ( e ) ≤ s to every edge e ∈ E ( G ) . For every vertex v ∈ V ( G ) the weighted degree is defined as: � wd ( v ) = w ( e ) . e ∋ v w is irregular if for v � = u we have wd ( v ) � = wd ( u ) . Irregularity strength s ( G ) : the lowest s that allows some irregular labeling. Introduced by G. Chartrand, M.S. Jacobson, J. Lehel, O.R. Oellermann, S. Ruiz, F. Saba, 1988. Marcin Anholcer Group Edge Irregularity Strength of Graphs 4/ 32
Notation Outline Irregularity Strength Introduction Edge Irregularity Strength Group Edge Irregularity Strength Labelling the Graph with Abelian Groups The End Group Irregularity Strength s ( G ) : Some results Lower bound: n i + i − 1 s ( G ) ≥ max i 1 ≤ i ≤ ∆ Best upper bound (M. Kalkowski, M. Karoński, F. Pfender, 2009): � 6 n � s ( G ) ≤ δ Exact values for some families of graphs (e.g. cycles, grids, some kinds of trees, circulant graphs). Marcin Anholcer Group Edge Irregularity Strength of Graphs 5/ 32
Notation Outline Irregularity Strength Introduction Edge Irregularity Strength Group Edge Irregularity Strength Labelling the Graph with Abelian Groups The End Group Irregularity Strength s ( G ) : Some results Lower bound: n i + i − 1 s ( G ) ≥ max i 1 ≤ i ≤ ∆ Best upper bound (M. Kalkowski, M. Karoński, F. Pfender, 2009): � 6 n � s ( G ) ≤ δ Exact values for some families of graphs (e.g. cycles, grids, some kinds of trees, circulant graphs). Marcin Anholcer Group Edge Irregularity Strength of Graphs 5/ 32
Notation Outline Irregularity Strength Introduction Edge Irregularity Strength Group Edge Irregularity Strength Labelling the Graph with Abelian Groups The End Group Irregularity Strength s ( G ) : Some results Lower bound: n i + i − 1 s ( G ) ≥ max i 1 ≤ i ≤ ∆ Best upper bound (M. Kalkowski, M. Karoński, F. Pfender, 2009): � 6 n � s ( G ) ≤ δ Exact values for some families of graphs (e.g. cycles, grids, some kinds of trees, circulant graphs). Marcin Anholcer Group Edge Irregularity Strength of Graphs 5/ 32
Notation Outline Irregularity Strength Introduction Edge Irregularity Strength Group Edge Irregularity Strength Labelling the Graph with Abelian Groups The End Group Irregularity Strength es ( G ) : Definition Assign positive integer w ( v ) ≤ s to every vertex v ∈ V ( G ) . For every edge e = uv ∈ E ( G ) the weight is defined as: wd ( uv ) = w ( u ) + w ( v ) . w is irregular if for every two edges e � = f we have wt ( e ) � = wt ( f ) . Edge Irregularity Strength es ( G ) : the lowest s that allows some irregular labeling. Introduced by A. Ahmad, O. Bin Saeed Al-Mushayt, M. Baˇ ca, 2014. Marcin Anholcer Group Edge Irregularity Strength of Graphs 6/ 32
Notation Outline Irregularity Strength Introduction Edge Irregularity Strength Group Edge Irregularity Strength Labelling the Graph with Abelian Groups The End Group Irregularity Strength es ( G ) : Definition Assign positive integer w ( v ) ≤ s to every vertex v ∈ V ( G ) . For every edge e = uv ∈ E ( G ) the weight is defined as: wd ( uv ) = w ( u ) + w ( v ) . w is irregular if for every two edges e � = f we have wt ( e ) � = wt ( f ) . Edge Irregularity Strength es ( G ) : the lowest s that allows some irregular labeling. Introduced by A. Ahmad, O. Bin Saeed Al-Mushayt, M. Baˇ ca, 2014. Marcin Anholcer Group Edge Irregularity Strength of Graphs 6/ 32
Notation Outline Irregularity Strength Introduction Edge Irregularity Strength Group Edge Irregularity Strength Labelling the Graph with Abelian Groups The End Group Irregularity Strength es ( G ) : Definition Assign positive integer w ( v ) ≤ s to every vertex v ∈ V ( G ) . For every edge e = uv ∈ E ( G ) the weight is defined as: wd ( uv ) = w ( u ) + w ( v ) . w is irregular if for every two edges e � = f we have wt ( e ) � = wt ( f ) . Edge Irregularity Strength es ( G ) : the lowest s that allows some irregular labeling. Introduced by A. Ahmad, O. Bin Saeed Al-Mushayt, M. Baˇ ca, 2014. Marcin Anholcer Group Edge Irregularity Strength of Graphs 6/ 32

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