A Note On Line Graphs

In this note we define two generalizations of the line graph and obtain some results. Also, we mark some open problems

Atom-bond-connectivity index of certain graphs

The ABC index is one of the most applicable topological graph indices and several properties of it has been studied already due to its extensive chemical applications. Several variants of it have also been defined and used for several reasons. In this paper, we calculate the atom-bond connectivity index of some derived graphs such as double graphs, subdivision graphs and complements of some standard graphs.Publisher's Versio

t-Path Sigraphs

[[abstract]]Given a sigraph S and a positive integer t, the t-path sigraph (S)(subscript t) of S is formed by taking a copy of the vertex set V (S) of S, joining two vertices u and v in the copy by a single edge e=uv whenever there is a u-v path of length t in S and then by defining its sign to be-whenever in every u-v path of length t in S all the edges are negative. In this paper, we introduce a variation of the concept of t-path sigraphs studied above. The motivation stems naturally from one's mathematically inquisitiveness as to ask why not define the sign of an edge e=uv in (S) (subscript t) as the product of the signs of the vertices u and v in S. It is shown that for any sigraph S, its t-path sigraph (S) (subscript t) is balanced. We then give structural characterization of t-path sigraphs. Further, in this paper we characterize sigraphs which are switching equivalent to their 2(3)-path sigraphs

S-Antipodal Signed Graphs

[[abstract]]In this paper we introduced a new notion S-antipodal signed graph of a signed graph and its properties are obtained. Also we give the re- lation between antipodal signed graphs and S-antipodal signed graphs. Further, we discuss structural characterization of S-antipodal signed graphs

Note on Path Signed Graphs

Abstract Data in the social sciences can often modeled using signed graphs, graphs where every edge has a sign + or −, or marked graphs, graphs where every vertex has a sign + or −. The path graph P k (G) of a graph G is obtained by representing the paths P k in G by vertices whenever the corresponding paths P k in G from a path P k+1 or a cycle C k . In this note, we introduce a natural extension of the notion of path graphs to the realm of signed graphs. It is shown that for any signed graph S, P k (S) is balanced. The concept of a line signed graph is generalized to that of a path signed graphs. Further, in this note we discuss the structural characterization of path signed graphs. Also, we characterize signed graphs which are switching equivalent to their path signed graphs P 3 (S) (P 4 (S)). 2000 Mathematics Subject Classification : 05C 2

A Note on Path Signed Digraphs

Particularly, a Smarandachely 2-signed digraph or Smarandachely 2-marked digraph is called abbreviated a signed digraph or a marked digraph