On Certain Topological Indices of Signed Graphs
aa r X i v : . [ m a t h . G M ] F e b ON CERTAIN TOPOLOGICAL INDICES OF SIGNED GRAPHS
SUDEV NADUVATH , ∗ , JOHAN KOK Department of Mathematics, CHRIST (Deemed to be University), Bangalore 560029, India.
Copyright c (cid:13)
Abstract.
The first Zagreb index of a graph G is the sum of squares of the vertex degrees in a graph and thesecond Zagreb index of G is the sum of products of degrees of adjacent vertices in G . The imbalance of an edgein G is the numerical difference of degrees of its end vertices and the irregularity of G is the sum of imbalances ofall its edges. In this paper, we extend the concepts of these topological indices for signed graphs and discuss thecorresponding results on signed graphs. Keywords:
Positive Zagreb indices; negative Zagreb indices; signed imbalance indices; net-imbalance; Zagrebnet-indices; Gutman indices. I NTRODUCTION
For general notation and concepts in graph theory, we refer to [4, 16, 21] and for the termi-nology in signed graphs, see [22, 23]. Unless mentioned otherwise, all graphs considered hereare finite, simple, undirected and connected.Let G ( V , E ) be a graph with vertex set V ( G ) = { v , v , . . . , v n } and edge set E ( G ) , where | E ( G ) | = m . The degree of a vertex v i in G is the number of edges incident on it and is denotedby d G ( v i ) . If the context is clear, let us use the notation d i instead of d G ( v i ) .The first Zagreb index , denoted by M ( G ) and the second Zagreb index , denoted by M ( G ) is defined in [13] as M ( G ) = ∑ v i ∈ V ( G ) d i and M ( G ) = n ∑ v i v j ∈ E ( G ) d i d j . The imbalance of an edge e = uv ∈ E ( G ) is defined as imb G ( uv ) = | d G ( u ) − d G ( v ) | (see [2]). The notion of irregularity of ∗ Corresponding authorE-mail address: [email protected] October 4, 2019 a graph G has also been introduced in [2] as irr ( G ) = ∑ uv ∈ E ( G ) imb ( uv ) . Another new measureof irregularity of a simple undirected graph G , called the total irregularity of G , denoted by irr t ( G ) , is defined in [1] as irr t ( G ) = ∑ u , v ∈ V ( G ) | d ( u ) − d ( v ) | .We extend the notions of these topological indices of graphs defined in the previous sectionto the theory of signed graphs.A signed graph (see [22, 23]), denoted by S ( G , σ ) , is a graph G ( V , E ) together with a function σ : E ( G ) → { + , −} that assigns a sign, either + or − , to each ordinary edge in G . The function σ is called the signature or sign function of S , which is defined on all edges except half edgesand is required to be positive on free loops. The unsigned graph G is called the underlyinggraph of the signed graph S .An edge e of S is said to be positive or negative in accordance with its signature σ ( e ) ispositive or negative. The number of positive edges incident on a vertex v in S is the positivedegree of v and is denoted by d + S ( v ) and the number of negative edges incident on v is the negative degree of v and is denoted by d − S ( v ) . Clearly, d G ( v ) = d + S ( v ) + d − S ( v ) .Analogous to the definition of first Zagreb index of a graph, we can define two types of firstZagreb indices for a given signed graph S as follows. Defiition 1.
Let S be a signed graph and let d + i and d − i be the positive and negative degree of avertex v i in S . Then, the first positive Zagreb index of S is denoted by M + ( S ) is defined as M + ( S ) = ∑ v i ∈ V ( G ) (cid:0) d + i (cid:1) , (1)the first negative Zagreb index of S is denoted by M − ( S ) is M − ( S ) = ∑ v i ∈ V ( G ) (cid:0) d − i (cid:1) (2)and the first mixed Zagreb index of a signed graph S is denoted by M ∗ and is defined as M ∗ ( S ) = ∑ v i ∈ V ( G ) d + i d − i (3)In a similar way, we can also define the second Zagreb indices for a signed graph S as follows. ERTAIN TOPOLOGICAL INDICES OF SIGNED GRAPHS 3
Defiition 2.
Let S be a signed graph and let d + i and d − i be the positive and negative degree of avertex v i in S . Then, the second positive Zagreb index of S is denoted by M + ( S ) is defined as M + ( S ) = ∑ v i v j ∈ E ( G ) d + i d + j ; 1 ≤ i = j ≤ n , (4)the second negative Zagreb index of S is denoted by M − ( S ) is M − ( S ) = ∑ v i v j ∈ E ( G ) d − i d − j ; 1 ≤ i = j ≤ n (5)and the second mixed Zagreb index of a signed graph S is denoted by M ∗ and is defined as M ∗ ( S ) = ∑ v i v j ∈ E ( G ) d + i d − j ; 1 ≤ i = j ≤ n . (6)In view of the new notions defined above, the relation between the first Zagreb indices of asigned graph S and the first Zagreb index of its underlying graph G is discussed in the followingtheorem. Theorem 1.
For a signed graph S and its underlying graph G,(i) M ( G ) = M + ( S ) + M − ( S ) + M ∗ ( S ) ,(ii) M ( G ) = M + ( S ) + M − ( S ) + M ∗ ( S ) .Proof. Let d i denotes the degree of a vertex v i in the underlying graph G of a signed graph S .Then, we have N. K. SUDEV AND J.KOK ( i ) M ( G ) = ∑ v i ∈ V ( G ) ( d i ) = ∑ v i ∈ V ( G ) ( d + i + d − i ) = ∑ v i ∈ V ( G ) ( d + i ) + ∑ v i ∈ V ( G ) ( d − i ) + ∑ v i ∈ V ( G ) d + i d − i = M + ( S ) + M − ( S ) + ∑ v i ∈ V ( G ) d + i d − i M ( G ) = M + ( S ) + M − ( S ) + M ∗ ( S ) . ( ii ) M ( G ) = ∑ v i v j ∈ E ( G ) d i d j = ∑ v i v j ∈ E ( G ) ( d + i + d − i )( d + j + d − j )= ∑ v i v j ∈ E ( G ) d + i d + j + ∑ v i v j ∈ E ( G ) d − i d − j + ∑ v i v j ∈ E ( G ) d + i d − j = M + ( S ) + M − ( S ) + M ∗ ( S ) . (cid:3) Analogous to the definition of imbalance of edges in graphs, let us introduce the followingdefinitions for signed graphs.
Defiition 3.
For an edge e = uv in a signed graph S , the positive imbalance of e can be definedas imb + S ( uv ) = | d + S ( u ) − d + S ( v ) | and the negative imbalance of e can be defined as imb − S ( uv ) = | d − S ( u ) − d − S ( v ) | .In a similar way, the two types irregularities of a signed graph can be defined as follows. Defiition 4.
The positive irregularity of a signed graph S , denoted by irr + ( S ) , is defined to be irr + ( S ) = ∑ uv ∈ E ( S ) imb + S ( uv ) and the negative irregularity of S , denoted by irr − ( S ) , is defined as irr − ( S ) = ∑ uv ∈ E ( S ) imb − S ( uv ) .The total irregularities of a signed graph S can also be defined as given below. ERTAIN TOPOLOGICAL INDICES OF SIGNED GRAPHS 5
Defiition 5.
The total positive irregularity of a signed graph S , denoted by irr + t ( S ) , is definedas irr + t ( G ) = ∑ u , v ∈ V ( S ) | d + S ( u ) − d + S ( v ) | . and the total negative irregularity of a signed graph S , denoted by irr + t ( S ) , is defined as irr − t ( G ) = ∑ u , v ∈ V ( S ) | d − S ( u ) − d − S ( v ) | . The following theorem discusses the relation between the irregularities and total irregularitiesof a signed graph S with the corresponding indices of its underlying graph G . Theorem 2.
Let S be a signed graph and G denotes its underlying graph. Then, we have(i) imb G ( uv ) ≤ imb + S ( uv ) + imb − S ( uv ) ; uv ∈ E ( S ) ( and E ( G )) ,(ii) irr ( G ) ≤ irr + ( S ) + irr − ( S ) ,(iii) irr t ( G ) ≤ irr + t ( S ) + irr − t ( S ) .Proof. Let S be a signed graph with underlying graph G and let e = uv be any edge of S (and G ). Then, imb G ( uv ) = | d G ( u ) − d G ( v ) | = | (cid:0) d + ( u ) + d − ( u ) (cid:1) − (cid:0) d + ( v ) + d − ( v ) (cid:1) | = | (cid:0) d + ( u ) − d + ( v ) (cid:1) + (cid:0) d − ( u ) − d − ( v ) (cid:1) |≤ | (cid:0) d + ( u ) − d + ( v ) (cid:1) | + | (cid:0) d − ( u ) − d − ( v ) (cid:1) | ∴ imb G ( uv ) ≤ imb + S ( uv ) + imb − S ( uv ) . Also, irr ( G ) = ∑ uv ∈ E ( G ) imb G ( uv ) ≤ ∑ uv ∈ E ( S ) (cid:0) imb + S ( uv ) + imb − S ( uv ) (cid:1) = ∑ uv ∈ E ( S ) imb + S ( uv ) + ∑ uv ∈ E ( S ) imb − S ( uv ) ∴ irr ( G ) ≤ irr + ( S ) + irr − ( S ) . N. K. SUDEV AND J.KOK and irr t ( G ) = ∑ u , v ∈ V ( G ) | d G ( u ) − d G ( v ) | = ∑ u , v ∈ V ( G ) (cid:12)(cid:12)(cid:0) d + S ( u ) + d − S ( u ) (cid:1) − (cid:0) d + S ( v ) + d − S ( v ) (cid:1)(cid:12)(cid:12)! = ∑ u , v ∈ V ( G ) (cid:12)(cid:12)(cid:0) d + S ( u ) − d + S ( v ) (cid:1) + (cid:0) d − S ( u ) + d − S ( v ) (cid:1)(cid:12)(cid:12)! ≤ ∑ u , v ∈ V ( G ) (cid:12)(cid:12)(cid:0) d + S ( u ) − d + S ( v ) (cid:1)(cid:12)(cid:12) + ∑ u , v ∈ V ( G ) (cid:12)(cid:12)(cid:0) d − S ( u ) − d − S ( v ) (cid:1)(cid:12)(cid:12) ∴ irr t ( G ) ≤ irr + t ( S ) + irr − t ( S ) . (cid:3) The net-degree of a signed graph S , denoted by d ± S , is defined in [15] as d ± S ( v ) = d + S ( v ) − d − S ( v ) . The signed graph S is said to be net-regular if every vertex of S has the same net-degree.Different from the notation used in [15], we use notation ˆ d S ( v ) represent the net-degree of avertex in a signed graph S .Invoking the notion of the net-degree of vertices in a signed graph S , we introduce the fol-lowing notions on S . Defiition 6.
Let S be a signed graph and let ˆ d i denotes the net-degree of a vertex in S . The firstZagreb net-index of the signed graph S is denoted by M ( S ) and is defined as M ( S ) = ∑ v i ∈ V ( S ) ˆ d i (7)and the second Zagreb net-index of S is denoted by M ( S ) is defined as M ( S ) = ∑ v i v j ∈ E ( S ) ˆ d i ˆ d j (8)In view of the above notions, we have the following theorems. Theorem 3.
Let S be a signed graph and G be its underlying graph. Then,(i) M ( G ) = M ( S ) + M ∗ ( S ) ,(ii) M ( G ) = M ( S ) + M ∗ ( S ) . ERTAIN TOPOLOGICAL INDICES OF SIGNED GRAPHS 7
Proof.
Let d i , d + i , d − i respectively represent the degree, positive degree and negative degree ofa vertex v i in S . Then, we have ˆ d i = d + i − d − i . Then, we have M ( S ) = ∑ v i ∈ V ( S ) ˆ d i = ∑ v i ∈ V ( S ) ( d + i − d − i ) = ∑ v i ∈ V ( S ) ( d + i ) + ∑ v i ∈ V ( S ) ( d − i ) − ∑ v i ∈ V ( S ) d + i d − i = M + ( S ) + M − ( S ) − M ∗ ( S )= ( M ( G ) − M ∗ ( S )) − M ∗ ( S )= M ( G ) − M ∗ ( S ) . Similarly, M ( S ) = ∑ v i v j ∈ E ( S ) ˆ d i ˆ d j = ∑ v i v j ∈ E ( S ) ( d + i − d − i )( d + j − d − j )= ∑ v i v j ∈ E ( S ) d + i d + j + ∑ v i v j ∈ E ( S ) d − i d − j − ∑ v i v j ∈ E ( S ) d + i d − j ; i = j = M + ( S ) + M − ( S ) − M ∗ ( S )= ( M ( G ) − M ∗ ( S )) − M ∗ ( S )= M ( G ) − M ∗ ( S ) . (cid:3) Analogous to the definition of imbalance and irregularities of signed graphs mentioned in theprevious section, we introduce the following notions.
Defiition 7.
For an edge e = uv in a signed graph S , the net-imbalance or simply the imbalance of e , denoted by imb S ( uv ) , is defined as imb S ( uv ) = | ˆ d S ( u ) − ˆ d S ( v ) | . Defiition 8.
The irregularity of a signed graph S , denoted by irr ( S ) , is defined to be irr ( S ) = ∑ uv ∈ E ( S ) imb S ( uv ) = ∑ uv ∈ E ( S ) | ˆ d S ( u ) − ˆ d S ( v ) | . (9) N. K. SUDEV AND J.KOK
Defiition 9.
The total irregularity of a signed graph S , denoted by irr t ( S ) , is defined as irr t ( G ) = ∑ u , v ∈ V ( S ) | ˆ d S ( u ) − ˆ d S ( v ) | (10)Note that if a signed graph S is net-regular, then ˆ d i = ˆ d j ; ∀ i = j and hence we have irr ( S ) = irr t ( S ) = Theorem 4.
For any signed graph S, we have(i) imb S ( v i v j ) ≥ imb + S ( v i v j ) − imb − S ( v i v j ) ,(ii) irr ( S ) ≥ irr + ( S ) − irr − ( S ) ,(iii) irr t ( S ) ≥ irr + t ( S ) − irr − t ( S ) .Proof. Let e = v i v j be an arbitrary edge in G . Then imb S ( v i v j ) = | ˆ d i − ˆ d j | = | ( d + i − d − i ) − ( d + j − d − j ) ≥ | d + i − d + j | − | d − i − d − j | = imb + S ( v i v j ) − imb − S ( v i v j ) i . e ., imb S ( v i v j ) ≥ imb + S ( S ) − imb − S ( S ) . Also, irr ( S ) = ∑ v i v j ∈ E ( S ) | ˆ d i − ˆ d j | = | ∑ v i v j ∈ E ( S ) ( d + i − d − i ) − ( d + j − d − j ) ≥ ∑ v i v j ∈ E ( S ) | d + i − d + j | − ∑ v i v j ∈ E ( S ) | d − i − d − j | = ∑ v i v j ∈ E ( S ) imb + S ( v i v j ) − ∑ v i v j ∈ E ( S ) imb − S ( v i v j ) i . e ., imb S ( v i v j ) ≥ imb + S ( S ) − imb − S ( S ) . ERTAIN TOPOLOGICAL INDICES OF SIGNED GRAPHS 9 and irr t ( S ) = ∑ v i , v j ∈ V ( S ) | ˆ d i − ˆ d j | = ∑ v i , v j ∈ V ( S ) | ( d + i − d − i ) − ( d + j − d − j ) | = ∑ v i , v j ∈ V ( S ) | ( d + i − d + j ) − ( d − i − d − j ) |≥ ∑ v i , v j ∈ V ( S ) | d + i − d + j | − ∑ v i , v j ∈ V ( S ) | d − i − d − j | = irr + t ( S ) − irr − t ( S ) i . e ., irr t ( S ) ≥ irr + t ( S ) − irr − t ( S ) . (cid:3) The
Schultz index of a graph G is defined as S ( G ) = ∑ u , v ∈ V ( d G ( u ) + d G ( v )) d G ( u , v ) (see [18]).Analogous to this terminology, we introduce the following notions for signed graphs. Defiition 10.
The positive Schultz index of a signed graph S , denoted by S + ( S ) , is defined to be S + ( S ) = ∑ u , v ∈ V ( S ) (cid:0) d + S ( u ) + d + S ( v ) (cid:1) d S ( u , v ) and the negative Schultz index of the signed graph S , denoted by S − ( S ) , is defined to be S − ( S ) = ∑ u , v ∈ V ( S ) (cid:0) d − S ( u ) + d − S ( v ) (cid:1) d S ( u , v ) . Here, note that the distance between two vertices in a signed graph S is the same as thedistance between those two vertices in the underlying graph G of S . In view of the abovenotions, we have the following theorem. Theorem 5.
If S be a signed graph and G be its underlying graph, then S ( G ) = S + ( S ) + S − ( S ) . Proof.
Let G be the underlying graph of a signed graph S . Then, for any two vertices u , v ∈ V ( S ) , d S ( u , v ) = d G ( u , v ) . Then, we have S ( G ) = ∑ u , v ∈ V ( S ) [ d G ( u ) + d G ( v )] d G ( u , v )= ∑ u , v ∈ V ( S ) (cid:2)(cid:0) d + S ( u ) + d − S ( u ) (cid:1) + (cid:0) d + S ( v ) + d − S ( v ) (cid:1)(cid:3) d S ( u , v )= ∑ u , v ∈ V ( S ) (cid:2)(cid:0) d + S ( u ) + d + S ( v ) (cid:1) + (cid:0) d − S ( u ) + d − S ( v ) (cid:1)(cid:3) d S ( u , v )= ∑ u , v ∈ V ( S ) (cid:2) d + S ( u ) + d + S ( v ) (cid:3) d S ( u , v ) + ∑ u , v ∈ V ( S ) (cid:2) d − S ( u ) + d − S ( v ) (cid:3) d S ( u , v )= S + ( S ) + S − ( S ) . (cid:3) Using the concepts of net-degree of vertices in a signed graph, we introduce the followingnotion.
Defiition 11.
The
Schultz index of a signed graph S , denoted by S ( S ) , is defined as S ( S ) = ∑ u , v ∈ V ( S ) (cid:0) ˆ d S ( u ) + ˆ d S ( v ) (cid:1) d S ( u , v ) , where ˆ d ( v ) is the net-degree of a vertex v ∈ V ( S ) .Invoking the above definition, we have the following theorem on the Schultz index of signedgraphs. Theorem 6.
For a signed graph S, then S ( S ) = S + ( S ) − S − ( S ) . ERTAIN TOPOLOGICAL INDICES OF SIGNED GRAPHS 11
Proof.
Let G be the underlying graph of a signed graph S . Then, as mentioned in the previoustheorem, for any two vertices u , v ∈ V ( S ) , d S ( u , v ) = d G ( u , v ) . Then, we have S ( S ) = ∑ u , v ∈ V ( S ) (cid:2) ˆ d S ( u ) + ˆ d S ( v ) (cid:3) d S ( u , v )= ∑ u , v ∈ V ( S ) (cid:2)(cid:0) d + S ( u ) − d − S ( u ) (cid:1) + (cid:0) d + S ( v ) − d − S ( v ) (cid:1)(cid:3) d S ( u , v )= ∑ u , v ∈ V ( S ) (cid:2)(cid:0) d + S ( u ) + d + S ( v ) (cid:1) − (cid:0) d − S ( u ) + d − S ( v ) (cid:1)(cid:3) d S ( u , v )= ∑ u , v ∈ V ( S ) (cid:2) d + S ( u ) + d + S ( v ) (cid:3) d S ( u , v ) − ∑ u , v ∈ V ( S ) (cid:2) d − S ( u ) + d − S ( v ) (cid:3) d S ( u , v )= S + ( S ) − S − ( S ) . (cid:3) If G is the underlying graph of a signed graph S , then we have S ( G ) ≥ S ( S ) . Moreover, wehave S ( G ) + S ( S ) = S + ( S ) and S ( G ) − S ( S ) = S − ( S ) .The Gutman index of a graph G is another interesting topological index, denoted by G ( G ) ,which is defined as G ( G ) = ∑ u , v ∈ V d G ( u ) d G ( v ) d G ( u , v ) (see [10]). Analogous to this terminology,we introduce the following notions for signed graphs. Defiition 12.
The positive Gutman index of a signed graph S , denoted by G + ( S ) , is defined tobe G + ( S ) = ∑ u , v ∈ V ( S ) d + S ( u ) d + S ( v ) d S ( u , v ) . The negative Gutman index of the signed graph S , denoted by G − ( S ) , is defined as G − ( S ) = ∑ u , v ∈ V ( S ) d − S ( u ) d − S ( v ) d S ( u , v ) . The mixed Gutman index of the signed graph S , denoted by G ∗ ( S ) , is defined as G − ( S ) = ∑ u , v ∈ V ( S ) d + S ( u ) d − S ( v ) d S ( u , v ) . The following theorem discusses the relation between these Gutman indices of signed graphsand the Gutman index of its underlying graph.
Theorem 7.
If G is the underlying graph of a signed graph S, then G ( G ) = G + ( S ) + G − ( S ) + G ∗ ( S ) .Proof. G ( G ) = ∑ u , v ∈ V ( S ) d G ( u ) d G ( v ) d G ( u , v )= ∑ u , v ∈ V ( S ) (cid:2)(cid:0) d + S ( u ) + d − S ( u ) (cid:1) (cid:0) d + S ( v ) + d − S ( v ) (cid:1)(cid:3) d S ( u , v )= ∑ u , v ∈ V ( S ) (cid:2) d + S ( u ) d + S ( v ) + d + S ( u ) d − S ( v ) + d − S ( u ) d + S ( v ) + d − S ( u ) d − S ( v ) (cid:3) d S ( u , v )= ∑ u , v ∈ V ( S ) d + S ( u ) d + S ( v ) d S ( u , v ) + ∑ u , v ∈ V ( S ) , u = v d + S ( u ) d − S ( v ) d S ( u , v ) + ∑ u , v ∈ V ( S ) d − S ( u ) d − S ( v ) d S ( u , v )= G + ( S ) + G ∗ ( S ) + G − ( S ) . (cid:3) Using the concepts of net-degree of vertices in a signed graph, we now introduce the follow-ing notion.
Defiition 13.
The
Gutman index of a signed graph S , denoted by G ( S ) , is defined as G ( S ) = ∑ u , v ∈ V ( S ) ˆ d S ( u ) ˆ d S ( v ) d S ( u , v ) , where ˆ d ( v ) is the net-degree of a vertex v ∈ V ( S ) .Invoking the above definition, we have the following theorem on the Gutman index of signedgraphs. Theorem 8.
For a signed graph S, then G ( S ) = G + ( S ) + G − ( S ) − S ∗ ( S ) . ERTAIN TOPOLOGICAL INDICES OF SIGNED GRAPHS 13
Proof. G ( S ) = ∑ u , v ∈ V ( S ) ˆ d S ( u ) ˆ d S ( v ) d S ( u , v )= ∑ u , v ∈ V ( S ) (cid:2)(cid:0) d + S ( u ) − d − S ( u ) (cid:1) (cid:0) d + S ( v ) − d − S ( v ) (cid:1)(cid:3) d S ( u , v )= ∑ u , v ∈ V ( S ) (cid:2) d + S ( u ) d + S ( v ) − d + S ( u ) d − S ( v ) − d − S ( u ) d + S ( v ) + d − S ( u ) d − S ( v ) (cid:3) d S ( u , v )= ∑ u , v ∈ V ( S ) d + S ( u ) d + S ( v ) d S ( u , v ) − ∑ u , v ∈ V ( S ) , u = v d + S ( u ) d − S ( v ) d S ( u , v ) + ∑ u , v ∈ V ( S ) d − S ( u ) d − S ( v ) d S ( u , v )= G + ( S ) − G ∗ ( S ) + G − ( S ) . (cid:3) Several problems in this area are still open. Determining various other topological indicesfor are yet to be determined. Investigations on various theoretical and practical applications ofthese indices of graphs and signed graphs are also promising.
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