Deepa Sinha

Some results on semi-total signed graphs

A signed graph (or sigraph in short) is an ordered pair S = (Su, σ), where Su is a graph G = (V,E), called the underlying graph of S and σ : E → {+,−} is a function from the edge set E of Su into the set {+,−}, called the signature of S. The ×-line sigraph of S denoted by L×(S) is a sigraph defined on the line graph L(S u) of the graph Su by assigning to each edge ef of L(Su), the product of signs of the adjacent edges e and f in S. In this paper, first we define semi-total line sigraph and semi-total point sigraph of a given sigraph and then characterize balance and consistency of semi-total line sigraph and semi-total point sigraph.

Sufficiency and Duality in Nonsmooth Multiobjective Programming Problem under Generalized Univex Functions

We consider a nonsmooth multiobjective programming problem where the functions involved are nondifferentiable. The class of univex functions is generalized to a far wider class of --V-type I univex functions. Then, through various nontrivial examples, we illustrate that the class introduced is new and extends several known classes existing in the literature. Based upon these generalized functions, Karush-Kuhn-Tucker type sufficient optimality conditions are established. Further, we derive weak, strong, converse, and strict converse duality theorems for Mond-Weir type multiobjective dual program.

Some Starlike and Convexity Properties of Sakaguchi Classes for Hypergeometric Functions

The objective of this paper is to give some characterizations for a (Gaussian) hypergeometric function to be in a subclass of Sakaguchi type functions. Two subclasses S(;t ) and T (;t ) of Sakaguchi type functions in the open unit disc U are also discussed.

A Characterization of Line Sigraphs

Abstract A signed graph (or in short, sigraph ) is an ordered pair S = ( S u , s ) where S u is a graph G = ( V , E ) called the underlying graph of S and s : E(S u ) → {+, -} is a function denned on the edge set E ( S u ) = E into set {+, −}, called a signing of G We let E + ( S ) = [e ∈ E ( G ): s ( e ) = +} and E − ( S ) = [ e ∈ E ( G ) : s ( e ) = -}. Then the set E ( S ) = E + ( S ) U E − (S) is called the edge set of S , the elements of E + ( S )( E − ( S )) are called positive (negative) edges in S. In this way a graph may be regarded as a sigraph in which all the edges are positive; hence we regard graphs as all-positive sigraph (all-negative sigraphs are denned similarly). A sigraph is said to be homogeneous if it is either all-positive or all-negative and heterogenous otherwise. For a sigraph S , its line sigraph whose vertex set V(L(S)) is the edge set E ( S ) = E ( S u ) of S and two vertices of L(S) are joined by a negative edge if and only if they correspond to adjacent negative edges in S. In this paper, we define a given sigraph S to be a line sigraph if there exists a sigraph H such that L(H) ≅ S(read as L(H) is isomorphic to S ). We then give the following structural characterization of line sigraphs, extending the well known characterization of line graphs. Theorem : A signed graph S is a line sigraph if and only if (i) S u is a line graph and (ii) If uv is a positive edge of S then either there is no negative edge incident at u or there is no negative edge incident at v.

Non-smooth multiobjective programming problem involving (d I - ρ - σ)-V-type I functions

In this paper, we consider a multiobjective optimisation problem where the objective and constraint functions involved are non-differentiable. We introduce a new class of functions namely (dI - ρ - σ)-V-type I functions and illustrate through non-trivial examples that the class introduced is non-empty. We then obtain sufficient optimality conditions under the newly introduced class of functions and derive various weak, strong, converse and strict converse duality results for Wolfe type and Mond-Weir type dual programmes in order to relate the efficient and weak efficient solutions of primal and dual problems.

Characterization of signed line digraphs

Abstract Given a signed digraph S = ( V ( D ) , A ( D ) , σ ) on a given digraph D = ( V , A ) called the underlying digraph of S , its signed line digraph L ( S ) is a signed digraph defined on the line digraph L ( D ) of D by defining an arc e f in it to be negative if and only if both the arcs e and f in S are negative and oriented in the same direction through their common vertex. In this paper, we define a given signed digraph S to be a signed line digraph if there exists a signed digraph H such that L ( H ) ≅ S (read as “ L ( H ) is isomorphic to S ”). We derive three structural characterizations of signed line digraphs, extending the well known characterization of line digraphs due to Hemminger (1972) [10] .

Generalized (---)-Type I Univex Functions in Multiobjective Optimization

A new class of generalized functions (𝑑-𝜌-𝜂-𝜃)-type I univex is introduced for a nonsmooth multiobjective programming problem. Based upon these generalized functions, sufficient optimality conditions are established. Weak, strong, converse, and strict converse duality theorems are also derived for Mond-Weir-type multiobjective dual program.