F-index of graphs based on four operations related to the lexicographic product
aa r X i v : . [ c s . D M ] J un F-INDEX OF GRAPHS BASED ON FOUROPERATIONS RELATED TO THELEXICOGRAPHIC PRODUCT
Nilanjan De ∗ Department of Basic Sciences and Humanities (Mathematics),Calcutta Institute of Engineering and Management, Kolkata, India.
Abstract
The forgotten topological index or F-index of a graph is defined as the sum ofcubes of the degree of all the vertices of the graph. In this paper we study theF-index of four operations related to the lexicographic product on graphs whichwere introduced by Sarala et al. [D. Sarala, H. Deng, S.K. Ayyaswamya and S.Balachandrana, The Zagreb indices of graphs based on four new operations relatedto the lexicographic product,
Applied Mathematics and Computation , (2017)156-169.].MSC(2010): Primary: 05C35; Secondary: 05C07, 05C40 Keywords:
Topological Index, Degree, Zagreb Index, F-Index, graph operations,lexicographic product
1. Introduction
Let, G = ( V , E ) be a connected, undirected simple graph with vertex set V = V ( G ) and edge set E = E ( G ). The degree of a vertex v in G is definedas the number of edges incident to v and denoted by d G ( v ). In chemical graph the-ory, chemical structures are considered as a graph, often called molecular graphand a molecular structure descriptor or topological index is a number obtained ∗ Corresponding Author.
Email address: [email protected] (Nilanjan De) rom a molecular graph and are structurally invariant. Generally, topological in-dices show a good correlation with di ff erent physico-chemical properties of corre-sponding chemical compounds, so that now a days topological indices are used asa standard tool in studying isomer discrimination and structure-property relationsfor predicting di ff erent properties of chemical compounds and biological activi-ties. Thus, topological indices has shown there applicability in chemistry, bio-chemistry, nanotechnology and even discovery and design of new drugs. Thereare various types of topological indices among which the first and second Zagrebindices are most important, most studied and have good correlations to di ff erentchemical properties vertex-degree based topological indices. These indices wereintroduced in 1972 [1], denoted by M ( G ) and M ( G ) and are respectively definedas M ( G ) = P v ∈ V ( G ) d G ( v ) = P uv ∈ E ( G ) [ d G ( u ) + d G ( v )] and M ( G ) = P uv ∈ E ( G ) d G ( u ) d G ( v ).These indices attracted more and more attention from chemists and mathe-maticians, specially for di ff erent graph operations [2, 3]. Another topologicalindex, named as “forgotten topological index” or ”F-index” [4] by Furtula andGutman is defined as sum of cubes of degrees of the vertices of the graph wasalso introduced in [1]. Furtula et al., in [5], investigate some basic properties andbounds of F-index and in [6] Abdoa et al. found the extremal trees with respect tothe F-index. Recently, the present author studied this index for di ff erent graph op-erations [7] and of di ff erent classes of nanostar dendrimers [9] and also introducedF-coindex in [8]. Also, the present author studied F-index of di ff erent transforma-tion graphs and four sum of graphs in [10] and [11] respectively. The F-index ofa graph G is denoted by F ( G ), so that F ( G ) = X v ∈ V ( G ) d G ( v ) = X uv ∈ E ( G ) [ d G ( u ) + d G ( v ) ] . (1)One of the redefined version of Zagreb index denoted by ReZ M ( G ) and isdefined as ReZ M ( G ) = X uv ∈ E ( G ) d G ( u ) d G ( v )[ d G ( u ) + d G ( v )] . (2)The general first Zagreb index of a graph G was introduced by Li et al. in [12]and is defined as ξ n ( G ) = X v ∈ V ( G ) d G ( v ) n = X uv ∈ E ( G ) [ d G ( u ) n − + d G ( v ) n − ] (3)2here n is an integer, not 0 or 1. Obviously ξ ( G ) = M ( G ) and ξ ( G ) = F ( G ).There are various subdivision related derived graphs of any graph G . For anyconnected graph G , the four derived graphs S ( G ), R ( G ), Q ( G ) and T ( G ) of G aredefined as follows:(a) The subdivision graph S ( G ) is obtained from G by adding a new vertexcorresponding to every edge of G , that is, each edge of G is replaced by a path oflength two.(b) The graph R ( G ) is obtained from G by adding a new vertex correspond-ing to every edge of G , then joining each new vertex to the end vertices of thecorresponding edge that is, each edge of G is replaced by a triangle.(c) The graph Q ( G ) is obtained from G by adding a new vertex correspondingto every edge of G , then joining with edges those pairs of new vertices on adjacentedges of G .(a) The total graph T ( G ) of a graph G has its vertices as the edges and ver-tices of G and adjacency in T ( G ) is defined by the adjacency or incidence of thecorresponding elements of G .For di ff erent properties and use of the these four derived graphs S ( G ), R ( G ), Q ( G ) and T ( G ) of G , we refer our reader to [13, 14, 15, 16].Considering the above four derived graphs, M. Eliasi and B. Taeri introducedfour new graph operations named as F-sum graphs in [17], which is based onCartesian product of graphs. There are various studies of these F-sum graphs inrecent literature [18, 19, 20, 21, 3, 11]. Another important type of graph operation,named as the composition or lexicographic product of two connected graphs G and G , denoted by G [ G ], is a graph such that the set of vertices is V ( G ) × V ( G )and two vertices u = ( u , v ) and v = ( u , v ) of G [ G ] are adjacent if andonly if either u is adjacent with u or u = u and v is adjacent with v .In [22], Sarala et al. introduced four new operations named as F-product, onthese subdivision related graphs based on lexicographic product of two connectedgraphs G and G as follows: Definition 1.
Let F = { S , R , Q , T } , then the F-product of G and G , denoted byG [ G ] F , is defined by F ( G )[ G ] − E ∗ , where E ∗ = { ( u , v )( u , v ) ∈ E ( F ( G )[ G ]) : u ∈ V ( F ( G )) − V ( G ) , v v ∈ E ( G ) } i.e., G [ G ] F is a graph with the set ofvertices V ( G [ G ] F ) = ( V ( G ) ∪ E ( G )) × V ( G ) and two vertices u = ( u , v ) and v = ( u , v ) of G [ G ] are adjacent if and only if either [ u = u ∈ V ( G ) andv v ∈ E ( G )] or [ u u ∈ E ( F ( G )) and v , v ∈ V ( G )] . In [22], Sarala et al. derived explicit expressions of first and second Zagreb indicesof F-product graphs. 3 . Main Results
In this section, if not indicated otherwise, for the graph G i , the notation V ( G i )and E ( G i ) are used for the vertex set and edge set respectively, whereas n i and m i denote the number of vertices and the number of edges of the graph G i , i ∈ { , } ,respectively. In the following we now derive explicit expressions of F-index ofthe graphs G [ G ] S , G [ G ] R , G [ G ] Q and G [ G ] T respectively. Theorem 1.
Let G and G be two connected graphs. ThenF ( G [ G ] S ) = n F ( G ) + n F ( G ) + n m M ( G ) + n m M ( G ) + n m . Proof.
Let, d ( u , v ) = d G [ G ] S ( u , v ) be the degree of any vertex ( u , v ) in the graph G [ G ] S . Then from definition of F-index of graph, we have F ( G [ G ] S ) = X ( u , v )( u , v ) ∈ E ( G [ G ] S ) [ d ( u , v ) + d ( u , v ) ] = X u = u ∈ V ( G ) X v v ∈ E ( G ) [ d ( u , v ) + d ( u , v ) ] + X v ∈ V ( G ) X v ∈ V ( G ) X u u ∈ E ( S ( G )) [ d ( u , v ) + d ( u , v ) ] = S + S Now, S = X u = u ∈ V ( G ) X v v ∈ E ( G ) [ d ( u , v ) + d ( u , v ) ] = X u ∈ V ( G ) X v v ∈ E ( G ) [ { n d G ( u ) + d G ( v ) } + { n d G ( u ) + d G ( v ) } ] = X u ∈ V ( G ) X v v ∈ E ( G ) [ n d G ( u ) + d G ( v ) + n d G ( u ) d G ( v ) + n d G ( u ) + d G ( v ) + n d G ( u ) d G ( v )] = X u ∈ V ( G ) X v v ∈ E ( G ) [2 n d G ( u ) + { d G ( v ) + d G ( v ) } + n d G ( u ) { d G ( v ) + d G ( v ) } ] = n m M ( G ) + n F ( G ) + n m M ( G ) . Again, S = X v ∈ V ( G ) X v ∈ V ( G ) X u u ∈ E ( S ( G )) [ d ( u , v ) + d ( u , v ) ]4 X v ∈ V ( G ) X v ∈ V ( G ) X u ∈ V ( G ) , a ∈ V ( G ) u and a are ad jacent [ d ( u , v ) + d ( a , v ) ] = X v ∈ V ( G ) X v ∈ V ( G ) X u ∈ V ( G ) , a ∈ V ( G ) u and a are ad jacent [ { n d G ( u ) + d G ( v ) } + { n } ] = X v ∈ V ( G ) X v ∈ V ( G ) X u ∈ V ( G ) d G ( u )[ n d G ( u ) + d G ( v ) + n d G ( u ) d G ( v ) + n ] = X v ∈ V ( G ) X v ∈ V ( G ) X u ∈ V ( G ) [ n d G ( u ) + d G ( u ) d G ( v ) + n d G ( u ) d G ( v ) + n d G ( u )] = n F ( G ) + n m M ( G ) + n m M ( G ) + n m . Combining, S and S , we get the desired result as theorem 1. (cid:3) Theorem 2.
Let G and G be two connected graphs. ThenF ( G [ G ] R ) = n F ( G ) + n F ( G ) + n m M ( G ) + n m M ( G ) + n m . Proof.
Let, d ( u , v ) = d G [ G ] R ( u , v ) be the degree of any vertex ( u , v ) in the graph G [ G ] R . Then similarly, from definition of F-index of graph, we have F ( G [ G ] R ) = X ( u , v )( u , v ) ∈ E ( G [ G ] R ) [ d ( u , v ) + d ( u , v ) ] = X u = u ∈ V ( G ) X v v ∈ E ( G ) [ d ( u , v ) + d ( u , v ) ] + X v ∈ V ( G ) X v ∈ V ( G ) X u u ∈ E ( R ( G )) [ d ( u , v ) + d ( u , v ) ] = R + R . Now, R = X u = u ∈ V ( G ) X v v ∈ E ( G ) [ d ( u , v ) + d ( u , v ) ] = X u = u ∈ V ( G ) X v v ∈ E ( G ) [ { n d R ( G ) ( u ) + d G ( v ) } + { n d R ( G ) ( u ) + d G ( v ) } ] = X u ∈ V ( G ) X v v ∈ E ( G ) [ { n d G ( u ) + d G ( v ) } + { n d G ( u ) + d G ( v ) } ] = X u ∈ V ( G ) X v v ∈ E ( G ) [4 n d G ( u ) + d G ( v ) + n d G ( u ) d G ( v )5 n d G ( u ) + d G ( v ) + n d G ( u ) d G ( v )] = X u ∈ V ( G ) X v v ∈ E ( G ) [8 n d G ( u ) + { d G ( v ) + d G ( v ) } + n d G ( u ) { d G ( v ) + d G ( v ) } ] = n m M ( G ) + n F ( G ) + n m M ( G ) . (4)Now, R = X v ∈ V ( G ) X v ∈ V ( G ) X u u ∈ E ( R ( G )) [ d ( u , v ) + d ( u , v ) ] = X v ∈ V ( G ) X v ∈ V ( G ) X u u ∈ E ( G ) [ d ( u , v ) + d ( u , v ) ] + X v ∈ V ( G ) X v ∈ V ( G ) X u u ∈ E ( R ( G )) u ∈ V ( G ) , u ∈ V ( R ( G )) − V ( G ) [ d ( u , v ) + d ( u , v ) ] = R ′ + R ′′ . Now, R ′ = X v ∈ V ( G ) X v ∈ V ( G ) X u u ∈ E ( G ) [ d ( u , v ) + d ( u , v ) ] = X v ∈ V ( G ) X v ∈ V ( G ) X u u ∈ E ( G ) [ { n d R ( G ) ( u ) + d G ( v ) } + { n d R ( G ) ( u ) + d G ( v ) } ] = X v ∈ V ( G ) X v ∈ V ( G ) X u u ∈ E ( G ) [ { n d G ( u ) + d G ( v ) } + { n d G ( u ) + d G ( v ) } ] = X v ∈ V ( G ) X v ∈ V ( G ) X u u ∈ E ( G ) [4 n d G ( u ) + d G ( v ) + n d G ( u ) d G ( v ) + n d G ( u ) + d G ( v ) + n d G ( u ) d G ( v )] = X v ∈ V ( G ) X v ∈ V ( G ) X u u ∈ E ( G ) [4 n { d G ( u ) + d G ( u ) } + { d G ( v ) + d G ( v ) } + n { d G ( u ) d G ( v ) + d G ( u ) d G ( v ) } ] = n F ( G ) + n m M ( G ) + n m M ( G ) . (5)Similarly, R ′′ = X v ∈ V ( G ) X v ∈ V ( G ) X u u ∈ E ( R ( G )) u ∈ V ( G ) , u ∈ V ( R ( G )) − V ( G ) [ d ( u , v ) + d ( u , v ) ]6 X v ∈ V ( G ) X v ∈ V ( G ) X u u ∈ E ( R ( G )) u ∈ V ( G ) , u ∈ V ( R ( G )) − V ( G ) [ { n d R ( G ) ( u ) + d G ( v ) } + { n d R ( G ) ( u ) } ] = X v ∈ V ( G ) X v ∈ V ( G ) X u u ∈ E ( R ( G )) u ∈ V ( G ) , u ∈ V ( R ( G )) − V ( G ) [ { n d G ( u ) + d G ( v ) } + { n } ] = X v ∈ V ( G ) X v ∈ V ( G ) X u u ∈ E ( R ( G )) u ∈ V ( G ) , u ∈ V ( R ( G )) − V ( G ) [4 n d G ( u ) + d G ( v ) + n d G ( u ) d G ( v ) + n ] = n X u ∈ V ( G ) d G ( u ) + n M ( G ) X u ∈ V ( G ) d G ( u ) + n m X u ∈ V ( G ) d G ( u ) + n X u ∈ V ( G ) d G ( u ) = n F ( G ) + n m M ( G ) + n m M ( G ) + n m . Hence combining the above results we get the desired result. (cid:3)
Theorem 3.
Let G and G be two connected graphs. ThenF ( G [ G ] Q ) = n F ( G ) − n F ( G ) + n ReZ M ( G ) + n H M ( G ) + n m M ( G ) + n m M ( G ) + n ξ ( G ) − n M ( G ) . Proof.
Let, d ( u , v ) = d G [ G ] Q ( u , v ) be the degree of any vertex ( u , v ) in the graph G [ G ] Q . So, from definition of F-index of graph, we can write F ( G [ G ] Q ) = X ( u , v )( u , v ) ∈ E ( G [ G ] Q ) [ d ( u , v ) + d ( u , v ) ] = X u = u ∈ V ( G ) X v v ∈ E ( G ) [ d ( u , v ) + d ( u , v ) ] + X v ∈ V ( G ) X v ∈ V ( G ) X u u ∈ E ( Q ( G )) [ d ( u , v ) + d ( u , v ) ] = Q + Q . Now, Q = X u = u ∈ V ( G ) X v v ∈ E ( G ) [ d ( u , v ) + d ( u , v ) ]7 X u ∈ V ( G ) X v v ∈ E ( G ) [ { n d Q ( G ) ( u ) + d G ( v ) } + { n d Q ( G ) ( u ) + d G ( v ) } ] = X u ∈ V ( G ) X v v ∈ E ( G ) [ n d G ( u ) + d G ( v ) + n d G ( u ) d G ( v ) + n d G ( u ) + d G ( v ) + n d G ( u ) d G ( v )] = X u ∈ V ( G ) X v v ∈ E ( G ) [2 n d G ( u ) + { d G ( v ) + d G ( v ) } + n d G ( u ) { d G ( v ) + d G ( v ) } ] = n m M ( G ) + n F ( G ) + n m M ( G ) . Again, Q = X v ∈ V ( G ) X v ∈ V ( G ) X u u ∈ E ( Q ( G )) [ d ( u , v ) + d ( u , v ) ] = X v ∈ V ( G ) X v ∈ V ( G ) X u u ∈ E ( Q ( G )) u ∈ V ( G ) , u ∈ V ( Q ( G )) − V ( G ) [ d ( u , v ) + d ( u , v ) ] + X v ∈ V ( G ) X v ∈ V ( G ) X u u ∈ E ( Q ( G )) u , u ∈ V ( Q ( G )) − V ( G ) [ d ( u , v ) + d ( u , v ) ] = Q ′ + Q ′′ . Now, Q ′ = X v ∈ V ( G ) X v ∈ V ( G ) X u u ∈ E ( Q ( G )) u ∈ V ( G ) , u ∈ V ( Q ( G )) − V ( G ) [ d ( u , v ) + d ( u , v ) ] = X v ∈ V ( G ) X v ∈ V ( G ) X u u ∈ E ( Q ( G )) u ∈ V ( G ) , u ∈ V ( Q ( G )) − V ( G ) [ { n d Q ( G ) ( u ) + d G ( v ) } + { n d Q ( G ) ( u ) } ] = X v ∈ V ( G ) X v ∈ V ( G ) X u u ∈ E ( Q ( G )) u ∈ V ( G ) , u ∈ V ( Q ( G )) − V ( G ) [ n d G ( u ) + d G ( v ) + n d G ( u ) d G ( v ) + n d G ( u ) ] = n F ( G ) + n m M ( G ) + n m M ( G ) + n X v ∈ V ( G ) X v ∈ V ( G ) X u u ∈ E ( Q ( G )) u ∈ V ( G ) , u ∈ V ( Q ( G )) − V ( G ) d Q ( G ) ( u ) . Now, since d Q ( G ) ( u ) = d G ( w i ) + d G ( w j ), for u ∈ V ( Q ( G )) − V ( G ), where8 is the vertex inserted into the edge w i w j of G , we have X u u ∈ E ( Q ( G )) u ∈ V ( G ) , u ∈ V ( Q ( G )) − V ( G ) d Q ( G ) ( u ) = X w i w j ∈ E ( G ) [ d G ( w i ) + d G ( w j )] = H M ( G ) (6)Thus, Q ′ = n F ( G ) + n m M ( G ) + n m M ( G ) + n H M ( G ).Again, Q ′′ = X v ∈ V ( G ) X v ∈ V ( G ) X u u ∈ E ( Q ( G )) u , u ∈ V ( Q ( G )) − V ( G ) [ d ( u , v ) + d ( u , v ) ] = X v ∈ V ( G ) X v ∈ V ( G ) X u u ∈ E ( Q ( G )) u , u ∈ V ( Q ( G )) − V ( G ) [ { n d Q ( G ) ( u ) } + { n d Q ( G ) ( u ) } ] = X v ∈ V ( G ) X v ∈ V ( G ) X u u ∈ E ( Q ( G )) u , u ∈ V ( Q ( G )) − V ( G ) n [ { d G ( w i ) + d G ( w j ) } + { d G ( w j ) + d G ( w k ) } ] = n [2 X w j ∈ V ( G ) C d G wj ) × d G ( w j ) + X w j ∈ V ( G ) ( d G ( w j ) − X w i ∈ V ( G ) , w i w j ∈ E ( G ) d G ( w i ) + X w j ∈ V ( G ) d G ( w j )( d G ( w j ) − X w i ∈ V ( G ) , w i w j ∈ E ( G ) d G ( w i )] = n [ X w j ∈ V ( G ) { d G ( w j ) − d G ( w j ) } + X w j ∈ V ( G ) ( d G ( w j ) − X w i ∈ V ( G ) , w i w j ∈ E ( G ) d G ( w i ) + X w j ∈ V ( G ) d G ( w j )( d G ( w j ) − X w i ∈ V ( G ) , w i w j ∈ E ( G ) d G ( w i )] = n [ ξ ( G ) − F ( G ) − M ( G ) + ReZ M ( G ) − F ( G )] . (7)Adding the above contributions we get the desired result as theorem 3. (cid:3) Theorem 4.
Let G and G be two connected graphs. ThenF ( G [ G ] T ) = n F ( G ) − n F ( G ) + n ReZ M ( G ) + n H M ( G ) + n m M ( G ) + n m M ( G ) + n ξ ( G ) − n M ( G ) . Proof.
We have, from definition of total graph T ( G ) d G [ G ] T ( u , v ) = d G [ G ] R ( u , v ) = n d G ( u ) + d G ( v ), for u ∈ V ( G ) and v ∈ V ( G ), 9 G [ G ] T ( u , v ) = d G [ G ] Q ( u , v ) = n d Q ( G ) ( u ), for u ∈ V ( T ( G )) − V ( G ) and v ∈ V ( G ).Then, from the definition of F-index, we have F ( G [ G ] T ) = X ( u , v )( u , v ) ∈ E ( G [ G ] T ) [ d G [ G ] T ( u , v ) + d G [ G ] T ( u , v ) ] = X u ∈ V ( G ) X v v ∈ E ( G ) [ d G [ G ] T ( u , v ) + d G [ G ] T ( u , v ) ] + X v ∈ V ( G ) X v ∈ V ( G ) X u u ∈ E ( T ( G )) [ d G [ G ] T ( u , v ) + d G [ G ] T ( u , v ) ] = X u ∈ V ( G ) X v v ∈ E ( G ) [ d G [ G ] R ( u , v ) + d G [ G ] R ( u , v ) ] + X v ∈ V ( G ) X v ∈ V ( G ) X u u ∈ E ( G ) [ d G [ G ] R ( u , v ) + d G [ G ] R ( u , v ) ] + X v ∈ V ( G ) X v ∈ V ( G ) X u u ∈ E ( T ( G )) u ∈ V ( G ) , u ∈ V ( T ( G )) − V ( G ) [ d G [ G ] R ( u , v ) + d G [ G ] Q ( u , v ) ] + X v ∈ V ( G ) X v ∈ V ( G ) X u u ∈ E ( T ( G )) u , u ∈ V ( T ( G )) − V ( G ) [ d G [ G ] Q ( u , v ) + d G [ G ] Q ( u , v ) ]Now, we have from (4), (5), (6) and (7) X u ∈ V ( G ) X v v ∈ E ( G ) [ d G [ G ] R ( u , v ) + d G [ G ] R ( u , v ) ] = n m M ( G ) + n F ( G ) + n m M ( G ) , and X v ∈ V ( G ) X v ∈ V ( G ) X u u ∈ E ( G ) [ d G [ G ] R ( u , v ) + d G [ G ] R ( u , v ) ] = n F ( G ) + n m M ( G ) + n m M ( G )and also X v ∈ V ( G ) X v ∈ V ( G ) X u u ∈ E ( T ( G )) u ∈ V ( G ) , u ∈ V ( T ( G )) − V ( G ) [ d G [ G ] R ( u , v ) + d G [ G ] Q ( u , v ) ] = X v ∈ V ( G ) X v ∈ V ( G ) X u u ∈ E ( T ( G )) u ∈ V ( G ) , u ∈ V ( T ( G )) − V ( G ) [ { n d G ( u ) + d G ( v ) } + { n d Q ( G ) ( u ) } ]10 X v ∈ V ( G ) X v ∈ V ( G ) X u u ∈ E ( T ( G )) u ∈ V ( G ) , u ∈ V ( T ( G )) − V ( G ) [4 n d G ( u ) + d G ( v ) + n d G ( u ) d G ( v ) + n d Q ( G ) ( u ) ] = n F ( G ) + n m M ( G ) + n m M ( G ) + n H M ( G ) . Finally X v ∈ V ( G ) X v ∈ V ( G ) X u u ∈ E ( T ( G )) u , u ∈ V ( T ( G )) − V ( G ) [ d G [ G ] Q ( u , v ) + d G [ G ] Q ( u , v ) ] = n [ ξ ( G ) − F ( G ) − M ( G ) + ReZ M ( G )] . Now adding the above contributions, we get the desired result. (cid:3)
Example 1.
Let G = P n and G = P m . Then applying Theorems 1-4, for thesegraphs with n = n, n = m, ( i ) F ( P n [ P m ] S ) = nm − m + nm − m + m − nm + m − n , ( ii ) F ( P n [ P m ] R ) = nm − m + nm − m − nm + m − nm + m − n , ( iii ) F ( P n [ P m ] Q ) = nm − m + nm − m + m − nm + m − n , ( iv ) F ( P n [ P m ] T ) = nm − m + nm − m − nm + m − nm + m − n .
3. Conclusion
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