In contrast to the canonically conjugate variates $q$,$p$ representing the
position and momentum of a particle in the phase space distributions, the three
Cartesian components, $J_{x}$,$J_{y}$, $J_{z}$ of a spin-$j$ system constitute
the mutually non-commuting variates in the quasi-probabilistic spin
distributions. It can be shown that a univariate spin distribution is never
squeezed and one needs to look into either bivariate or trivariate
distributions for signatures of squeezing. Several such distributions result if
one considers different characteristic functions or moments based on various
correspondence rules. As an example, discrete probability distribution for an
arbitrary spin-1 assembly is constructed using Wigner-Weyl and Margenau-Hill
correspondence rules. It is also shown that a trivariate spin-1 assembly
resulting from the exposure of nucleus with non-zero quadrupole moment to
combined electric quadrupole field and dipole magnetic field exhibits squeezing
in cerain cases.Comment: 13 pages, 1 Table, Presented at ICSSUR-05, Franc

Channu, B. C.

Dass, C.

Eregowda, G. B.

Kalpana, H. N.

Ramesh, L.

Thimmaiah, K. N.

English

arXiv

Cerium(IV) sulfate oxidizes 10-[3'-[N-bis(hydroxyethyl)amino]propyl]-2-chlorophenoxazine [BPCP] reversibly to a pink-colored radical cation [BPCP+·] in the presence of stoichiometric amounts (BPCP:Ce(IV) = 1:1) of the reactants. The radical cation underwent a second one-electron oxidation to form a brownish yellow-colored dication [BPCP2+] in the presence of more than one equivalent of Ce(IV), which was characterized by IR, mass-spectral and UV-vis spectroscopy. The cyclic voltammogram of BPCP exhibited two reversible anodic waves at 640 mV and 1057 mV and two cathodic waves at 582 mV and 930 mV at a scan rate of 24 mV/s. The peak at 640 mV corresponds to the oxidation of BPCP to the radical cation [BPCP+·] and the second anodic peak at 1057 mV stands for the oxidation of the radical cation to dication [BPCP2+]. Bromine oxidized BPCP to three products, as evidenced by HPLC. The tentatively predicted structures based on the mass-spectral data support the formation of brominated oxidized products. The respective first and second formal potentials of BPCP were found to be 782 mV and 936 mV and the transition potential of BPCP in the tritration of ascorbic acid with N-bromosuccinimide was found to be 787 mV in 0.5 M sulfuric acid. The optimum conditions for the use of BPCP as a redox indicator in the macro and micro determination of ascorbic acid, methionine, isoniazid, phenylhydrazine hydrochloride and biotin using N-bromosuccinimide as an oxidant have been developed

Dass, Chhabil

University of Mysore - Digital Repository of Research, Innovation and Scholarship (ePrints@UoM)

Southeast AsianBulletin ofMathematicsc⃝SEAMS. 2010Southeast Asian Bulletin of Mathematics (2010) 34: 1077–1082The Edge 푪4 Signed Graph of a Signed GraphR. RangarajanDepartment of Studies in Mathematics, University of Mysore, Soldevanahalli, IndiaEmail: rajra63@gmail.comP. Siva Kota ReddyDepartment of Mathematics, Acharya Institute of Technology, Bangalore-560 090,IndiaEmail: pskreddy@acharya.ac.inReceived 29 January 2009Accepted 9 June 2009Communicated by M.K. SenAMS Mathematics Subject Classification(2000): 05C22Abstract. A signed graph (marked graph) is an ordered pair 푆 = (퐺, 휎) (푆 = (퐺,휇)),where 퐺 = (푉,퐸) is a graph called the underlying graph of 푆 and 휎 : 퐸 → {+,−}(휇 : 푉 → {+,−}) is a function. The edge 퐶4 graph 퐸4(퐺) of a graph 퐺 = (푉,퐸) hasall edges of 퐺 as it vertices, two vertices in 퐸4(퐺) are adjacent if their correspondingedges in 퐺 are either incident or are opposite edges of some 퐶4. Analogously, one candefine the the edge 퐶4 signed graph of a signed graph 푆 = (퐺, 휎) as a signed graph퐸4(푆) = (퐸4(퐺), 휎′), where 퐸4(퐺) is the underlying graph of 퐸4(푆), where for anyedge 푒1푒2 in 퐸4(푆), 휎′(푒1푒2) = 휎(푒1)휎(푒2). It is shown that for any signed graph 푆, itsedge 퐶4 signed graph 퐸4(푆) is balanced. We then give structural characterization ofedge 퐶4 signed graphs. Two signed graphs 푆1 and 푆2 are switching equivalent written푆1 ∼ 푆2, whenever there exists a marking 휇 of 푆1 such that the signed graph 풮휇(푆1)obtained by changing the sign of every edge of 푆1 to its opposite whenever its endvertices are of opposite signs, is isomorphic to 푆2. Further, we obtain a structuralcharacterization of signed graphs that are switching equivalent to their edge 퐶4 signedgraphs.Keywords: Signed graphs; Balance; Switching; Edge 퐶4 signed graph; Negation of asigned graph.1078 R. Rangarajan and P.S.K. Reddy1. IntroductionFor graph theoretical terminologies and notations in this paper we follow thebook [4]; the non-standard will be given in this paper as and when required. Wetreat only finite simple graphs without self loops and isolates.A signed graph is an ordered pair 푆 = (퐺, 휎), where 퐺 = (푉,퐸) is a graphcalled underlying graph of S and 휎 : 퐸 → {+,−} is a function. A signed graph푆 = (퐺, 휎) is balanced if every cycle in 푆 has an even number of negative edges(See [2]). Equivalently a signed graph is balanced if product of signs of the edgeson every cycle of 푆 is positive. A marking of 푆 is a function 휇 : 푉 (퐺)→ {+,−};A signed graph 푆 together with a marking 휇 is denoted by 푆휇. In a signed graph푆 = (퐺, 휎), for any 퐴 ⊆ 퐸(퐺) the sign 휎(퐴) is the product of the signs on theedges of 퐴.The following characterization of balanced signed graphs is well known.Proposition 1.1. [9] A signed graph 푆 = (퐺, 휎) is balanced if and only if thereexist a marking 휇 of its vertices such that each edge 푢푣 in 푆 satisfies 휎(푢푣) =휇(푢)휇(푣).The idea of switching a signed graph was introduced by Abelson and Rosen-berg [1] in connection with structural analysis of marking 휇 of a signed graph 푆.Switching 푆 with respect to a marking 휇 is the operation of changing the sign ofevery edge of 푆 to its opposite whenever its end vertices are of opposite signs. Thesigned graph obtained in this way is denoted by 풮휇(푆) and is called 휇-switchedsigned graph or just switched signed graph. Two signed graphs 푆1 = (퐺, 휎)and 푆2 = (퐺′, 휎′) are said to be isomorphic, written as 푆1 ∼= 푆2 if there existsa graph isomorphism 푓 : 퐺 → 퐺′ (that is a bijection 푓 : 푉 (퐺) → 푉 (퐺′) suchthat if 푢푣 is an edge in 퐺 then 푓(푢)푓(푣) is an edge in 퐺′) such that for anyedge 푒 ∈ 퐺, 휎(푒) = 휎′(푓(푒)). Further a signed graph 푆1 = (퐺, 휎) switches to asigned graph 푆2 = (퐺′, 휎′) (or that 푆1 and 푆2 are switching equivalent) written푆1 ∼ 푆2, whenever there exists a marking 휇 of 푆1 such that 풮휇(푆1) ∼= 푆2. Notethat 푆1 ∼ 푆2 implies that 퐺 ∼= 퐺′, since the definition of switching does notinvolve change of adjacencies in the underlying graphs of the respective signedgraphs. In [11], the authors extended the above notion ‘switching’ to signeddirected graphs (see also, E. Sampathkumar et al. [10]).Two signed graphs 푆1 = (퐺, 휎) and 푆2 = (퐺′, 휎′) are said to be weaklyisomorphic (see [12]) or cycle isomorphic (see [13]) if there exists an isomorphism휙 : 퐺→ 퐺′ such that the sign of every cycle 푍 in 푆1 equals to the sign of 휙(푍)in 푆2. The following result is well known (See [13]):Proposition 1.2. [13] Two signed graphs 푆1 and 푆2 with the same underlyinggraph are switching equivalent if and only if they are cycle isomorphic.2. Edge 푪4 Signed Graph of a Signed GraphThe edge 퐶4 graph 퐸4(퐺) of a graph 퐺 is defined in [8] as follows:The edgeThe Edge 퐶4 Signed Graph of a Signed Graph 1079퐶4 graph of a graph 퐺 = (푉,퐸) is a graph 퐸4(퐺) = (푉′, 퐸′), with vertexset 푉 ′ = 퐸(퐺) such that two vertices 푒 and 푓 are adjacent if and only if thecorresponding edges in 퐺 are either incident or opposite edges of some 퐶4. Sotwo vertices are adjacent vertices in 퐸4(퐺) if the union of the correspondingedges in 퐺 induces any one of the graphs 푃3, 퐶3, 퐶4,퐾4 − {푒},퐾4.Clearly the edge 퐶4 graph coincides with the line graph for any acyclic graph.But they differ in many properties. In [7], the authors proved that the edge 퐶4graph 퐸4(퐺) has no forbidden subgraphs.In this paper, we extend the notion of 퐸4(퐺) to realm of signed graphs: Givena signed graph 푆 = (퐺, 휎) its edge 퐶4 signed graph 퐸4(푆) = (퐸4(퐺), 휎′) is thatsigned graph whose underlying graph is 퐸4(퐺), the edge 퐶4 graph of 퐺, wherefor any edge 푒1푒2 in 퐸4(푆), 휎′(푒1푒2) = 휎(푒1)휎(푒2).Hence, we shall call a given signed graph an edge 퐶4 signed graph if thereexists a signed graph 푆′ such that 푆 ∼= 퐸4(푆′). In the following subsection, weshall present a characterization of edge 퐶4 signed graphs.The following result indicates the limitations of the notion of edge 퐶4 signedgraphs as introduced above, since the entire class of unbalanced signed graphsis forbidden to be edge 퐶4 signed graphs.Proposition 2.1. For any signed graph 푆 = (퐺, 휎), its edge 퐶4 signed graph퐸4(푆) is balanced.Proof. Let 휎′denote the signing of 퐸4(푆) and let the signing 휎 of 푆 be treatedas a marking of the vertices of 퐸4(푆). Then by definition of 퐸4(푆) we see that휎′(푒1푒2) = 휎(푒1)휎(푒2), for every edge 푒1푒2 of 퐸4(푆) and hence, by Proposition1.1, the result follows.For any positive integer 푘, the 푘푡ℎ iterated edge 퐶4 signed graph, 퐸푘4 (푆) of푆 is defined as follows:퐸04(푆) = 푆, 퐸푘4 (푆) = 퐸4(퐸푘−14(푆)).Corollary 2.2. For any signed graph 푆 = (퐺, 휎) and any positive integer 푘,퐸푘4 (푆) is balanced.Recall the edge 퐶4 graph coincides with the line graph for any acyclic graph. Asa case, for a connected graph 퐺, 퐸4(퐺) = 퐺 if and only if 퐺 = 퐶푛, 푛 ∕= 4 (see[8]).We now characterize signed graphs that are switching equivalent to their edge퐶4 signed graphs.Proposition 2.3. For any signed graph 푆 = (퐺, 휎), 푆 ∼ 퐸4(푆) if and only if퐺 ∼= 퐶푛, where 푛 ∕= 4 and 푆 is balanced.1080 R. Rangarajan and P.S.K. ReddyProof. Suppose 푆 ∼ 퐸4(푆). This implies, 퐺 ∼= 퐸4(퐺) and hence by the aboveobservation we see that the graph 퐺 must be isomorphic to 퐶푛, where 푛 ∕= 4.Now, if 푆 is signed graph on 퐶푛, Proposition 2.1 implies that 퐸4(푆) is balancedand hence if 푆 is unbalanced its 퐸4(푆) being balanced cannot be switching equiv-alent to 푆 in accordance with Proposition 1.2. Therefore, 푆 must be balanced.Conversely, suppose that 푆 is balanced signed graph on 퐶푛, where 푛 ∕= 4.Then, since 퐸4(푆) is balanced as per Proposition 2.1 and since 퐺 ∼= 퐶푛, where푛 ∕= 4, the result follows from Proposition 1.2 again.The notion of negation 휂(푆) of a given signed graph 푆 defined in [5] as follows:휂(푆) has the same underlying graph as that of 푆 with the sign of each edgeopposite to that given to it in 푆. However, this definition does not say anythingabout what to do with nonadjacent pairs of vertices in 푆 while applying theunary operator 휂(.) of taking the negation of 푆.Proposition 2.3 provides easy solutions to two other signed graph switchingequivalence relations, which are given in the following results.Corollary 2.4. For any signed graph 푆 = (퐺, 휎), 휂(푆) ∼ 퐸4(푆) if and only if퐺 ∼= 퐶푛, where 푛 ≥ 5 and 푆 is unbalanced.Corollary 2.5. For any signed graph 푆 = (퐺, 휎), 퐸4(휂(푆)) ∼ 퐸4(푆).For a signed graph 푆 = (퐺, 휎), the 퐸4(푆) is balanced (Proposition 2.1). Wenow examine, the conditions under which negation 휂(푆) of 퐸4(푆) is balanced.Proposition 2.6. Let 푆 = (퐺, 휎) be a signed graph. If 퐸4(퐺) is bipartite then휂(퐸4(푆)) is balanced.Proof. Since, by Proposition 2.1, 퐸4(푆) is balanced, if each cycle 퐶 in 퐸4(푆)contains even number of negative edges. Also, since 퐸4(퐺) is bipartite, all cycleshave even length; thus, the number of positive edges on any cycle 퐶 in 퐸4(푆) isalso even. Hence 휂(퐸4(푆)) is balanced.In [7], the authors proved that for a connected complete multipartite graph퐺, 퐸4(퐺) is complete.The following result follows from the above observation and Proposition 2.1.Proposition 2.7. For a connected signed graph 푆 = (퐺, 휎), 퐸4(푆) is completebalanced signed graph if and only if 퐺 is complete multipartite graph.Proposition 2.8. [7] For a connected graph 퐺, 퐸4(퐺) is bipartite if and only if퐺 is either a path or 퐶2푛, 푛 ≥ 3.The Edge 퐶4 Signed Graph of a Signed Graph 1081From the above Proposition we have the following result:Proposition 2.9. For a connected signed graph 푆 = (퐺, 휎), 퐸4(푆) is bipartitebalanced signed graph if and only if 퐺 is isomorphic to either path or 퐶2푛, 푛 ≥ 3.Proof. Suppose 퐸4(푆) is bipartite balanced signed graph. This implies 퐸4(퐺)is bipartite and hence by Proposition 2.8 we see that the graph 퐺 must beisomorphic to either a path or 퐶2푛, 푛 ≥ 3.Conversely, suppose that 푆 is a signed graph on path or 퐶2푛, 푛 ≥ 3. Then,since 퐸4(푆) is balanced as per Proposition 2.1 and since 퐸4(퐺) is bipartite byProposition 2.8.3. Characterization of Edge 푪4 Signed GraphsThe following result characterize signed graphs which are edge 퐶4 signed graphs.Proposition 3.1. A signed graph 푆 = (퐺, 휎) is an edge 퐶4 signed graph if andonly if 푆 is balanced signed graph and its underlying graph 퐺 is an edge 퐶4 graph.Proof. Suppose that 푆 is balanced and 퐺 is an edge 퐶4 graph. Then there existsa graph 퐻 such that 퐸4(퐻) ∼= 퐺. Since 푆 is balanced, by Proposition 1.1, thereexists a marking 휇 of 퐺 such that each edge 푢푣 in 푆 satisfies 휎(푢푣) = 휇(푢)휇(푣).Now consider the signed graph 푆′ = (퐻,휎′), where for any edge 푒 in 퐻 , 휎′(푒) isthe marking of the corresponding vertex in 퐺. Then clearly, 퐸4(푆′) ∼= 푆. Hence푆 is an edge 퐶4 signed graph.Conversely, suppose that 푆 = (퐺, 휎) is an edge 퐶4 signed graph. Then thereexists a signed graph 푆′ = (퐻,휎′) such that 퐸4(푆′) ∼= 푆. Hence 퐺 is the edge퐶4 graph of 퐻 and by Proposition 2.1, 푆 is balanced.The reader may refer the papers [5, 6] for some more related topics in con-cerning the minimum number of vertices and edges in a graph with a given vertexconecticity, edge conectivity and mimimm degree. We investigate these notionsof 퐸4(푆) in a separate paper.Acknowledgement. The authors are grateful to the referee for his valuable sug-gestions for the improvement of the paper. Also, the second author very muchthankful to Sri. B. Premnath Reddy, Chairman, Acharya Institutes, for hisconstant support and encouragement for R & D.References[1] R.P. Abelson and M.J. Rosenberg, Symoblic psychologic: A model of attitudinalcognition, Behav. Sci. 3 (1958) 1–13.1082 R. Rangarajan and P.S.K. Reddy[2] F. Harary, On the notion of balance of a signed graph, Michigan Math. J. 2 (1953)143–146.[3] F. Harary, Structural duality, Behav. Sci. 2 (4) (1957) 255–265.[4] F. Harary, Graph Theory, Addison-Wesley Publishing Co., 1969.[5] F. Harary and J.A. Kabell, Extremal graphs with given connectivities, SoutheastAsain Bull. Math. 2 (2) (1978) 101–102.[6] K.M. Koh, Some problems on graph theory, Southeast Asain Bull. Math. 1 (1)(1977) 39–43.[7] M.K. Menon and A. Vijayakumar, The edge 퐶4 graph of a graph, In: Proc. ofICDM (2006), Ramanujan Math. Soc., Lecture Notes Series, Number 7, 2008.[8] E. Prisner, Graph Dyanamics, Longman, 1995.[9] E. Sampathkumar, Point signed and line signed graphs, Nat. Acad. Science. Let-ters 7 (3) (1984) 91–93.[10] E. Sampathkumar, P.S.K. Reddy, M.S. Subramanya, The line 푛-sigraph of a sym-metric 푛-sigraph, Southeast Asain Bull. Math. 34 (5) (2010) 953–958.[11] E. Sampathkumar, M.S. Subramanya and P.S.K. Reddy, Characterization of linesidigraphs, Southeast Asain Bull. Math., to appear.[12] T. Soz푎´nsky, Enueration of weak isomorphism classes of signed graphs, J. GraphTheory 4 (2) (1980) 127–144.[13] T. Zaslavsky, Signed graphs, Discrete Appl. Math. 4 (1) (1982) 47–74.

Characterization of Oxidized Products of 10-[3'-[N-Bis(hydroxyethyl)amino]propyl]-2-chlorophenoxazine and Its Applications in Titrations Involving N-Bromosuccinimide

Cerium(IV) sulfate oxidizes 10-3'-N-bis(hydroxyethyl)amino]propyl] BPCP] reversibly to a pink-colored radical cation BPCP+.] in the presence of stoichiometric amounts (BPCP:Ce(IV) = 1:1) of the reactants. The radical cation underwent a second one-electron oxidation to form a brownish yellow-colored dication BPCP2+] in the presence of more than one equivalent of Ce(IV), which was characterized by IR, mass-spectral and UV-vis spectroscopy. The cyclic voltammogram of BPCP exhibited two reversible anodic waves at 640 mV and 1057 mV and two cathodic waves at 582 mV and 930 mV at a scan rate of 24 mV/s. The peak at 640 mV corresponds to the oxidation of BPCP to the radical cation BPCP+.] and the second anodic peak at 1057 mV stands for the oxidation of the radical cation to dication BPCP2+]. Bromine oxidized BPCP to three products, as evidenced by HPLC. The tentatively predicted structures based on the mass-spectral data support the formation of brominated oxidized products. The respective first and second formal potentials of BPCP were found to be 782 mV and 936 mV and the transition potential of BPCP in the tritration of ascorbic acid with N-bromosuccinimide was found to be 787 mV in 0.5 M sulfuric acid. The optimum conditions for the use of BPCP as a redox indicator in the macro and micro determination of ascorbic acid, methionine, isoniazid, phenylhydrazine hydrochloride and biotin using N-bromosuccinimide as an oxidant have been developed

Characterization of Oxidized Products of 10-[3′-[N-Bis(hydroxyethyl)amino]propyl]-2-chlorophenoxazine and Its Applications in Titrations Involving N-Bromosuccinimide

Cerium(IV) sulfate oxidizes 10-3'-N-bis(hydroxyethyl)amino]propyl]-2-chlorophenoxazine BPCP reversibly to a pink-colored radical cation BPCP+Â· in the presence of stoichiometric amounts (BPCP:Ce(IV) = 1:1) of the reactants. The radical cation underwent a second one-electron oxidation to form a brownish yellow-colored dication BPCP2+ in the presence of more than one equivalent of Ce(IV), which was characterized by IR, mass-spectral and UV-vis spectroscopy. The cyclic voltammogram of BPCP exhibited two reversible anodic waves at 640 mV and 1057 mV and two cathodic waves at 582 mV and 930 mV at a scan rate of 24 mV/s. The peak at 640 mV corresponds to the oxidation of BPCP to the radical cation BPCP+Â· and the second anodic peak at 1057 mV stands for the oxidation of the radical cation to dication BPCP2+. Bromine oxidized BPCP to three products, as evidenced by HPLC. The tentatively predicted structures based on the mass-spectral data support the formation of brominated oxidized products. The respective first and second formal potentials of BPCP were found to be 782 mV and 936 mV and the transition potential of BPCP in the tritration of ascorbic acid with N-bromosuccinimide was found to be 787 mV in 0.5 M sulfuric acid. The optimum conditions for the use of BPCP as a redox indicator in the macro and micro determination of ascorbic acid, methionine, isoniazid, phenylhydrazine hydrochloride and biotin using N-bromosuccinimide as an oxidant have been developed

Channu, B.C.

Kalpana, H.N.

Eregowda, G.B.

Thimmaiah, K.N.

ePrints@Bangalore University

Characterization of oxidized products of 10-3'-N-bis(hydroxyethyl)amino]propyl]-2-chlorophenoxazine and its applications in titrations involving N-bromosuccinimide

In contrast to the canonically conjugate variates q, p representing the position and momentum of a particle in the phase space distributions, the three Cartesian components, Jx, Jy, Jz of a spin-j system constitute the mutually non-commuting variates in the quasi-probabilistic spin distributions. It can be shown that a univariate spin distribution is never squeezed and one needs to look into either bivariate or trivariate distributions for signatures of squeezing. Several such distributions result if one considers different characteristic functions or moments based on various correspondence rules. As an example, discrete probability distribution for an arbitrary spin-1 assembly is constructed using Wigner-Weyl and Margenau-Hill correspondence rules. It is also shown that a trivariate spin-1 assembly resulting from the exposure of nucleus with non-zero quadrupole moment to combined electric quadrupole field and dipole magnetic field exhibits squeezing in certain cases

Sirsi, Swarnamala