Mukti Acharya

Graceful signed graphs

A (p, q)-sigraph S is an ordered pair (G, s) where G = (V, E) is a (p, q)-graph and s is a function which assigns to each edge of G a positive or a negative sign. Let the sets E+ and E− consist of m positive and n negative edges of G, respectively, where m + n = q. Given positive integers k and d, S is said to be (k, d)-graceful if the vertices of G can be labeled with distinct integers from the set {0, 1, ..., k + (q − 1)d such that when each edge uv of G is assigned the product of its sign and the absolute difference of the integers assigned to u and v the edges in E+ and E− are labeled k, k + d, k + 2d, ..., k + (m − 1)d and −k, − (k + d), − (k + 2d), ..., − (k + (n − 1)d), respectively.In this paper, we report results of our preliminary investigation on the above new notion, which indeed generalises the well-known concept of (k, d)-graceful graphs due to B. D. Acharya and S. M. Hegde.

Boolean Petri Nets

Petri net is a graphical tool invented by Carl Adam Petri [13]. These are used for describing, designing and studying discrete event-driven dynamical systems that are characterized as being concurrent, asynchronous, distributed, parallel, random and/or nondeterministic. As a graphical tool, Petri net can be used for planning and designing a system with given objectives, more practically effective than flowcharts and block diagrams. As a mathematical tool, it enables one to set up state equations, algebraic equations and other mathematical models which govern the behavior of discrete dynamical systems. Still, there is a drawback inherent in representing discrete event-systems. They suffer from the state explosion problem as what will happen when a system is highly populated, i.e., initial state consists of a large number of places that are nonempty. This phenomenon may lead to an exponential growth of its reachability graph. This makes us to study the safe systems. The aim of this chapter is to present some basic results on 1-safe Petri nets that generate the elements of a Boolean hypercube as marking vectors. Complete Boolean hypercube is the most popular interconnection network with many attractive and well known properties such as regularity, symmetry, strong connectivity, embeddability, recursive construction, etc. For brevity, we shall call a 1-safe Petri net that generates all the binary n-vectors as marking vectors a Boolean Petri net. Boolean Petri nets are not only of theoretical interest but also are of practical importance, required in practice to construct control systems [1]. In this chapter, we will consider the problems of characterizing the class of Boolean Petri nets as also the class of crisp Boolean Petri nets, viz., the Boolean Petri nets that generate all the binary n-vectors exactly once. We show the existence of a disconnected Boolean Petri net whose reachability tree is homomorphic to the n-dimensional complete lattice Ln. Finally, we observe that characterizing a Boolean Petri net is rather intricate.

Construction of a Crisp Boolean Petri Net from a 1-safe Petri Net

The concept of a Petri net, a tool for the study of certain discrete dynamical systems, was invented in 1939 by Carl Adam Petri. In the attempt to characterize Boolean Petri nets, we discovered a subclass of Boolean Petri net called the crisp Boolean Petri net, viz., the one that generates every binary vector as its marking vectors exactly once. In this paper, the construction of a crisp Boolean Petri net from a 1-safe Petri net has been shown.

A Disconnected 1-Safe Petri Net Whose Reachability Tree Is Homomorphic to a Complete Boolean Lattice

Petri nets generating all the 2^n binary n-vectors as their marking vectors are not only of theoretical interest but also are of practical importance. In this note, we demonstrate the existence of a disconnected 1-safe Petri net whose reachability tree is homomorphic to the n-dimensional complete lattice L_n. This makes the problem of characterizing the 1-safe Petri nets that generate all the binary n-vectors as marking vectors exactly once appear more intricate.

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.

C -Cordial Labeling in Signed Graphs

Abstract In this paper, we initiate the study on C-cordial labeling in signed graphs and characterize paths and cycles with given number of negative sections and stars which admit C-cordial labeling.

Results on Lict Signed Graphs Lc(S)

Abstract A signed graph (or, in short, sigraph) S=(Su, α) consists of an underlying graph Su := G = (V, E) and a function a : E(Su) → {+, −}, called the signature of S. The line signed graph of a signed graph S, denoted L(S), is the signed graph having vertex set E(S) in which two of these vertices are adjacent if the corresponding edges are adjacent in S, every such edge ee’ in L(S) is negative whenever both the adjacent edges e and e’ in S are negative. The lict signed graph of a signed graph S, denoted Lc(S), is the signed graph having vertex set E(S) ∪ C(S) in which two of these vertices are adjacent if the corresponding members of S are adjacent or incident, every such edge uv in Lc(S) is negative whenever u, v ∈ E−(S) or u ∈ E−(S) and v is a cut-vertex of S with negative degree odd, here C(S) and E−(S) are cut-vertex set and negative edge set of S respectively. In this paper, we characterize signed graphs on Kp, p ≥ 2, on cycle Cn and on Km,n which are lict signed graphs or line signed graphs, characterize signed graphs S so that Lc(S) and L(S) are balanced. We also establish the characterization of signed graphs S for which S ∼ Lc(S), S ∼ L(S), h(S) ∼ Lc(S) and h(S) ∼ L(S), here h(S) is negation of S and ∼ stands for switching equivalence.

Zero Ring Labeling of Graphs

Abstract This paper introduces the notion of zero ring labeling of a graph and its empirical study demonstrates that every graph admits a zero ring labeling with respect to some zero ring. The zero ring graph Γ ( R 0 ) turns out to be maximal with respect to an injective zero ring labeling. In particular, we determine the optimal zero ring index for some well-known graphs.

Finite Abelian Group Labeling

Abstract In this paper, we introduce an abelian group labeling (shortly, AGL) over finite abelian groups. We have shown that every finite graph admits an abelian group labeling. In the course of investigation, we found that representation labeling can be obtained from abelian group labeling for certain graphs. Several new directions for further research are also indicated through problems.

Embedding an Arbitrary 1-safe Petri Net into a Boolean Petri Net

Petri nets are powerful mathematical formalism for designing and studying behaviors of a wide range of discrete dynamical event driven systems. The aim of this paper is to show that an arbitrary1safe Petri net can be embedded as an induced subnet of a Boolean Petri net, viz., the one that generates every binary n-vector as its marking vector.