An inequality between the edge-Wiener index and the Wiener index of - PowerPoint PPT Presentation

Introduction Lower bound Upper bound Ratio An inequality between the edge-Wiener index and the Wiener index of a graph A. Tepeh joint work with M. Knor and R. ˇ Skrekovski

Introduction Lower bound Upper bound Ratio Topological indices • derived from molecular graphs • numerical values

Introduction Lower bound Upper bound Ratio The Wiener index, defined as the sum of distances between all unordered pairs of vertices in a graph, is one of the most popular molecular descriptors.

Introduction Lower bound Upper bound Ratio The Wiener index, defined as the sum of distances between all unordered pairs of vertices in a graph, is one of the most popular molecular descriptors. • introduced by H. Wiener, 1947 • boiling point of paraffines is in strong correlation with the graph structure of their molecules • applications in chemistry, communication, facility location, cryptology, architecture,...

Introduction Lower bound Upper bound Ratio The Wiener index, defined as the sum of distances between all unordered pairs of vertices in a graph, is one of the most popular molecular descriptors. • introduced by H. Wiener, 1947 • boiling point of paraffines is in strong correlation with the graph structure of their molecules • applications in chemistry, communication, facility location, cryptology, architecture,... Our goal was to • compare Wiener index with the edge-Wiener index (to improve known results) • improve the upper bound for the edge-Wiener index • explore the ratio between both indices (find extremal graphs)

Introduction Lower bound Upper bound Ratio Basic definitions Let L ( G ) denote the line graph of G : V ( L ( G )) = E ( G ) and two distinct edges e , f ∈ E ( G ) adjacent in L ( G ) whenever they share an end-vertex in G G L(G) b a b a e c e c d d

Introduction Lower bound Upper bound Ratio Basic definitions • distance between vertices : d G ( u , v ) denotes the distance (=the length of a shortest path) between vertices u , v ∈ V ( G ) • distance between edges : d G ( e , f ) = d L ( G ) ( e , f ), � e = u 1 u 2 , f = v 1 v 2 if e � = f , then d ( e , f ) = min { d ( u i , v j ) : i , j ∈ { 1 , 2 }} + 1, if e = f , d ( e , f ) = 0 G L(G) b a b y a d(x,y)=3 e c d(e,b)=2 e c d(a,b)=1 x d d

Introduction Lower bound Upper bound Ratio Wiener index � W ( G ) = d ( u , v ) { u , v }⊆ V ( G ) edge-Wiener index � W e ( G ) = d ( e , f ) { e , f }⊆ E ( G ) • W e ( G ) = W ( L ( G )) • sometimes in the literature slightly different definition: � n � W e ( G ) + 2

Introduction Lower bound Upper bound Ratio • deg ( u ) = the degree of u ∈ V ( G ) • δ ( G ) = min { deg ( v ) : v ∈ V ( G ) } Gutman index � Gut ( G ) = deg ( u ) deg ( v ) d ( u , v ) { u , v }⊆ V ( G )

Introduction Lower bound Upper bound Ratio some known results Wu, 2010 • Let G be a connected graph of order n with δ ( G ) ≥ 2. Then W e ( G ) ≥ W ( G ) with equality if and only if G ∼ = C n .

Introduction Lower bound Upper bound Ratio some known results Wu, 2010 • Let G be a connected graph of order n with δ ( G ) ≥ 2. Then W e ( G ) ≥ W ( G ) with equality if and only if G ∼ = C n . • Let G be a connected graph of size m . Then 1 4( Gut ( G ) − m ) ≤ W e ( G ) ≤ 1 � m � 4( Gut ( G ) − m ) + . 2

Introduction Lower bound Upper bound Ratio • κ m ( G ) = the number of m -cliques in G cnik and ˇ Knor, Potoˇ Skrekovski, 2014 • Let G be a connected graph. Then W e ( G ) ≥ 1 4 Gut ( G ) − 1 4 | E ( G ) | + 3 4 κ 3 ( G ) + 3 κ 4 ( G ) (1) with equality in (1) if and only if G is a tree or a complete graph.

Introduction Lower bound Upper bound Ratio • κ m ( G ) = the number of m -cliques in G cnik and ˇ Knor, Potoˇ Skrekovski, 2014 • Let G be a connected graph. Then W e ( G ) ≥ 1 4 Gut ( G ) − 1 4 | E ( G ) | + 3 4 κ 3 ( G ) + 3 κ 4 ( G ) (1) with equality in (1) if and only if G is a tree or a complete graph. • Let G be a connected graph of minimal degree δ ≥ 2. Then W ( L ( G )) ≥ δ 2 − 1 W ( G ) . 4 • conjecture: W ( L ( G )) ≥ δ 2 4 W ( G )

Introduction Lower bound Upper bound Ratio main theorem Theorem Let G be a connected graph of minimum degree δ . Then, W e ( G ) ≥ δ 2 4 W ( G ) with equality holding if and only if G is isomorphic to a path on three vertices or a cycle.

Introduction Lower bound Upper bound Ratio For the proof we need... average distance of endpoints of edges e = u 1 u 2 and f = v 1 v 2 s ( u 1 u 2 , v 1 v 2 ) = 1 � � d ( u 1 , v 1 ) + d ( u 1 , v 2 ) + d ( u 2 , v 1 ) + d ( u 2 , v 2 ) 4

Introduction Lower bound Upper bound Ratio For the proof we need... average distance of endpoints of edges e = u 1 u 2 and f = v 1 v 2 s ( u 1 u 2 , v 1 v 2 ) = 1 � � d ( u 1 , v 1 ) + d ( u 1 , v 2 ) + d ( u 2 , v 1 ) + d ( u 2 , v 2 ) 4 Lemma Let G be a connected graph. Then s ( e , f ) = 1 � � � Gut ( G ) − | E ( G ) | . 4 { e , f }⊆ E ( G )

Introduction Lower bound Upper bound Ratio Lemma (Knor et al.,2014) Let u 1 u 2 , v 1 v 2 be a pair of edges of a connected graph G. Then d ( u 1 u 2 , v 1 v 2 ) ≥ s ( u 1 u 2 , v 1 v 2 ) + D ( u 1 u 2 , v 1 v 2 ) , (2) where − 1  if u 1 u 2 = v 1 v 2 ; 2   1 if the pair u 1 u 2 , v 1 v 2 forms a triangle ;   4 D ( u 1 u 2 , v 1 v 2 ) = 1 if the pair u 1 u 2 , v 1 v 2 forms a K 4 ;     0 otherwise. Moreover, equality holds in (2) if and only if ( i ) u 1 u 2 = v 1 v 2 , or ( ii ) the pair u 1 u 2 , v 1 v 2 forms a triangle or K 4 , or ( iii ) if u 1 u 2 and v 1 v 2 lie on a straight line.

Introduction Lower bound Upper bound Ratio • e , f ∈ E ( G ) • D ( e , f ) = d ( e , f ) − s ( e , f ) • if D ( e , f ) = α , we say that e , f forms a pair of type D α or that the pair e , f belongs to the set D α • if e = f , then D ( e , f ) = − 1 2 • I = { 0 , 1 4 , 1 2 , 3 4 , 1 }

Introduction Lower bound Upper bound Ratio • e , f ∈ E ( G ) • D ( e , f ) = d ( e , f ) − s ( e , f ) • if D ( e , f ) = α , we say that e , f forms a pair of type D α or that the pair e , f belongs to the set D α • if e = f , then D ( e , f ) = − 1 2 • I = { 0 , 1 4 , 1 2 , 3 4 , 1 } Lemma In a connected graph, every pair of distinct edges belongs to D α for some α ∈ I .

Introduction Lower bound Upper bound Ratio All types of pairs of two edges D D 3 D 1 1 4 4 u 1 k k k v 1 k k k k k+1 k+1 u 2 v 2 k k+1 k+1 D D ' D '' 1 1 0 2 2 k k k k k+1 k+1 k+1 k+1 k+1 k+1 k k+1 k+2 k k+1

Introduction Lower bound Upper bound Ratio � W e ( G ) = d ( e , f ) { e , f }⊆ E ( G ) � � = s ( e , f ) + D ( e , f ) { e , f }⊆ E ( G ) { e , f }⊆ E ( G )

Introduction Lower bound Upper bound Ratio � W e ( G ) = d ( e , f ) { e , f }⊆ E ( G ) � � = s ( e , f ) + D ( e , f ) { e , f }⊆ E ( G ) { e , f }⊆ E ( G ) Gut ( G ) − | E ( G ) | � = + D ( e , f ) 4 4 { e , f }⊆ E ( G )

Introduction Lower bound Upper bound Ratio � W e ( G ) = d ( e , f ) { e , f }⊆ E ( G ) � � = s ( e , f ) + D ( e , f ) { e , f }⊆ E ( G ) { e , f }⊆ E ( G ) Gut ( G ) − | E ( G ) | � = + D ( e , f ) 4 4 { e , f }⊆ E ( G ) Proposition Let G be a connected graph. Then W e ( G ) = Gut ( G ) − | E ( G ) | + | D 1 | + 1 4 | + 1 2 | + 3 4 | D 1 2 | D 1 4 | D 3 4 | . 4 4

Introduction Lower bound Upper bound Ratio Case 1: G is non-regular G has a vertex w ∈ V ( G ) of degree at least δ + 1. By previous proposition: 4 W e ( G ) = Gut ( G ) − | E ( G ) | + 4 | D 1 | + | D 1 4 | + 2 | D 1 2 | + 3 | D 3 4 | ≥ Gut ( G ) − | E ( G ) |

Introduction Lower bound Upper bound Ratio Case 1: G is non-regular G has a vertex w ∈ V ( G ) of degree at least δ + 1. By previous proposition: 4 W e ( G ) = Gut ( G ) − | E ( G ) | + 4 | D 1 | + | D 1 4 | + 2 | D 1 2 | + 3 | D 3 4 | ≥ Gut ( G ) − | E ( G ) | � = deg ( u ) deg ( v ) d ( u , v ) − | E ( G ) | { u , v }⊆ V ( G )

Introduction Lower bound Upper bound Ratio Case 1: G is non-regular G has a vertex w ∈ V ( G ) of degree at least δ + 1. By previous proposition: 4 W e ( G ) = Gut ( G ) − | E ( G ) | + 4 | D 1 | + | D 1 4 | + 2 | D 1 2 | + 3 | D 3 4 | ≥ Gut ( G ) − | E ( G ) | � = deg ( u ) deg ( v ) d ( u , v ) − | E ( G ) | { u , v }⊆ V ( G ) δ 2 � ≥ d ( u , v )+ { u , v }∈ V ( G ) \{ w } � deg ( u ) d ( u , w ) − | E ( G ) | ( δ + 1) u ∈ V ( G ) \{ w }

Introduction Lower bound Upper bound Ratio Case 1: G is non-regular G has a vertex w ∈ V ( G ) of degree at least δ + 1. By previous proposition: 4 W e ( G ) = Gut ( G ) − | E ( G ) | + 4 | D 1 | + | D 1 4 | + 2 | D 1 2 | + 3 | D 3 4 | ≥ Gut ( G ) − | E ( G ) | � = deg ( u ) deg ( v ) d ( u , v ) − | E ( G ) | { u , v }⊆ V ( G ) δ 2 � ≥ d ( u , v )+ { u , v }∈ V ( G ) \{ w } � deg ( u ) d ( u , w ) − | E ( G ) | ( δ + 1) u ∈ V ( G ) \{ w } � δ 2 W ( G ) + ≥ deg ( u ) − | E ( G ) | u ∈ V ( G ) \{ w }