On Certain Topological Indices of Signed Graphs

The first Zagreb index of a graph $G$ is the sum of squares of the vertex degrees in a graph and the second Zagreb index of $G$ is the sum of products of degrees of adjacent vertices in $G$. The imbalance of an edge in $G$ is the numerical difference of degrees of its end vertices and the irregularity of $G$ is the sum of imbalances of all its edges. In this paper, we extend the concepts of these topological indices for signed graphs and discuss the corresponding results on signed graphs.


INTRODUCTION
For general notation and concepts in graph theory, we refer to [4,16,21] and for the terminology in signed graphs, see [22,23]. Unless mentioned otherwise, all graphs considered here are finite, simple, undirected and connected.
Let G(V, E) be a graph with vertex set V (G) = {v 1 , v 2 , . . ., 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 denoted by 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 1 (G) and the second Zagreb index, denoted by M 2 (G) is defined in [13] as M 1 (G) = ∑ v i ∈V (G) 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 a graph G has also been introduced in [2] as irr(G) = ∑ uv∈E (G) imb (uv). Another new measure of 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) = 1 We extend the notions of these topological indices of graphs defined in the previous section to 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 edges and is required to be positive on free loops. The unsigned graph G is called the underlying graph of the signed graph S.
An edge e of S is said to be positive or negative in accordance with its signature σ (e) is positive or negative. The number of positive edges incident on a vertex v in S is the positive degree 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 . Analogous to the definition of first Zagreb index of a graph, we can define two types of first Zagreb indices for a given signed graph S as follows.
In view of the new notions defined above, the relation between the first Zagreb indices of a signed graph S and the first Zagreb index of its underlying graph G is discussed in the following theorem.
Theorem 1. For a signed graph S and its underlying graph G, 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 Analogous to the definition of imbalance of edges in graphs, let us introduce the following definitions for signed graphs.
Defiition 3. For an edge e = uv in a signed graph S, the positive imbalance of e can be defined In a similar way, the two types irregularities of a signed graph can be defined as follows. imb + S (uv) and the negative irregularity of S, denoted by irr − (S), is defined as The total irregularities of a signed graph S can also be defined as given below.
Defiition 5. The total positive irregularity of a signed graph S, denoted by irr + t (S), is defined as and the total negative irregularity of a signed graph S, denoted by irr + t (S), is defined as The following theorem discusses the relation between the irregularities and total irregularities of 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 Proof. Let S be a signed graph with underlying graph G and let e = uv be any edge of S (and G). Then, Also, The net-degree of a signed graph S, denoted by d ± S , is defined in [15] as d ± 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 notationd S (v) represent the net-degree of a vertex in a signed graph S.
Invoking the notion of the net-degree of vertices in a signed graph S, we introduce the following notions on S. Defiition 6. Let S be a signed graph and letd i denotes the net-degree of a vertex in S. The first Zagreb net-index of the signed graph S is denoted by M 1 (S) and is defined as and the second Zagreb net-index of S is denoted by M 2 (S) is defined as 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, Proof. Let d i , d + i , d − i respectively represent the degree, positive degree and negative degree of a vertex v i in S. Then, we haved Similarly, Analogous to the definition of imbalance and irregularities of signed graphs mentioned in the previous section, we introduce the following notions.
Defiition 9. The total irregularity of a signed graph S, denoted by irr t (S), is defined as Note that if a signed graph Sis net-regular, thend i =d j ; ∀ i = j and hence we have irr(S) = 0 and irr t (S) = 0.
Theorem 4. For any signed graph S, we have Proof. Let e = v i v j be an arbitrary edge in G. Then Also, [18]).
Analogous to this terminology, we introduce the following notions for signed graphs.
and the negative Schultz index of the signed graph S, denoted by S − (S), is defined to be .
Here, note that the distance between two vertices in a signed graph S is the same as the distance between those two vertices in the underlying graph G of S. In view of the above notions, 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), ). Then, we have Using the concepts of net-degree of vertices in a signed graph, we introduce the following notion.
Defiition 11. The Schultz index of a signed graph S, denoted by S(S), is defined as , whered(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 signed graphs.
Proof. Let G be the underlying graph of a signed graph S. Then, as mentioned in the previous theorem, for any two vertices u, v ∈ V (S), d S (u, v) = d G (u, v). Then, we have If G is the underlying graph of a signed graph S, then we have S(G) ≥ S(S). Moreover, we have S(G) + S(S) = 2S + (S) and S(G) − S(S) = 2S − (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.
The negative Gutman index of the signed graph S, denoted by G − (S), is defined as The mixed Gutman index of the signed graph S, denoted by G * (S), is defined as The following theorem discusses the relation between these Gutman indices of signed graphs and the Gutman index of its underlying graph.