Vertex-independent sets and vertex coverings as also edge-independent sets and edge coverings of graphs occur very naturally in many practical situations and hence have several potential applications. In this chapter, we study the properties of these sets. In addition, we discuss matchings in graphs and, in particular, in bipartite graphs. Matchings in bipartite graphs have varied applications in operations research. We also present two celebrated theorems of graph theory, namely, Tutte's 1-factor theorem and Hall's matching theorem. All graphs considered in this chapter are loopless. Vertex-Independent Sets and Vertex CoveringsDefinition 5.2.1. A subset S of the vertex set V of a graph G is called independent if no two vertices of S are adjacent in G. S Â V is a maximum independent set of G if G has no independent set S 0 with jS 0 j > jS j. A maximal independent set of G is an independent set that is not a proper subset of another independent set of G.For example, in the graph of Fig. 5.1, fu; v; wg is a maximum independent set and fx; yg is a maximal independent set that is not maximum. Definition 5.2.2.A subset K of V is called a covering of G if every edge of G is incident with at least one vertex of K. A covering K is minimum if there is no covering K 0 of G such that jK 0 j < jKjI it is minimal if there is no covering K 1 of G such that K 1 is a proper subset of K.In the graph W 5 of Fig. 5.2, fv 1 ; v 2 ; v 3 ; v 4 ; v 5 g is a covering of W 5 and fv 1 ; v 3 ; v 4 ; v 6 g is a minimal covering. Also, the set fx; yg is a minimum covering of the graph of Fig. 5.1.

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Thermoelectric properties of alkaline-earth-metal disilicides are strongly dependent on their electronic band structure in the vicinity of the Fermi level. In particular, the strontium disilicide, SrSi2 with a narrow band gap of about few tens of meV is composed of non-toxic, naturally abundant elements, and its thermoelectric properties are very sensitive to the substitution/alloying with third elements. In this article, we summarize the thermoelectric performance of substituted and Sr-deficient/Sr-rich SrSi2 alloys to realize the high thermoelectric figure-of-merit (ZT) for practical applications in the electronic and thermoelectric aspects, and also to explore the alternative routes to further improve its ZT value.

Background: Body Dysmorphic Disorder (BDD) is a psychiatric disorder with delusions about defects in appearance for which patients seek various treatments. Patients with BDD often seek cosmetic procedures, and orthodontic treatment is one among them. This is the first Indian study to determine the prevalence of BDD in an orthodontic outpatient department. Materials and method: A total of 1184 patients with varying degrees of malocclusion completed the BDD-YBOCS questionnaire, while an experienced orthodontist assessed the severity of malocclusion with a rating scale. Results: Sixty-two patients (5.2%) were screened positive for BDD. Most of the BDD-positive patients were single (p value of 0.02) and had multiple previous consultations for orthodontic treatment (p value of < 0.00**) with a gender predilection toward males (p value of 0.00**), and age was not statistically significant with a p value of 0.3. Conclusion: From our study, the prevalence of BDD among orthodontic patients was 5.2%. The orthodontist should be aware of the high prevalence of BDD among orthodontic patients and identify the expectations of the patient at the time of history taking and refer the patient to a psychiatrist for diagnosis and appropriate management.

A graph $G$ is said to have a parity-linked orientation $\phi$ if every even cycle $C_{2k}$ in $G^{\phi}$ is evenly (resp. oddly) oriented whenever $k$ is even (resp. odd). In this paper, this concept is used to provide an affirmative answer to the following conjecture of D. Cui and Y. Hou [D. Cui, Y. Hou, On the skew spectra of Cartesian products of graphs, Electronic J. Combin. 20(2):#P19, 2013]: Let $G=G(X,Y)$ be a bipartite graph. Call the $X\rightarrow Y$ orientation of $G,$ the canonical orientation. Let $\phi$ be any orientation of $G$ and let $Sp_S(G^{\phi})$ and $Sp(G)$ denote respectively the skew spectrum of $G^{\phi}$ and the spectrum of $G.$ Then $Sp_S(G^{\phi}) = {\bf{i}} Sp(G)$ if and only if $\phi$ is switching-equivalent to the canonical orientation of $G.$ Using this result, we determine the switch for a special family of oriented hypercubes $Q_d^{\phi},$ $d\geq 1.$ Moreover, we give an orientation of the Cartesian product of a bipartite graph and a graph, and then determine the skew spectrum of the resulting oriented product graph, which generalizes a result of Cui and Hou. Further this can be used to construct new families of oriented graphs with maximum skew energy.

Combined density functional theoretical (DFT) and ab initio methods have been used for the calculation of 13 C NMR chemical shifts of some hydrogen-terminated oligomers of ethylene, propylene, isobutylene, ethylene oxide, vinyl alcohol, and acrylonitrile. The 13 C isotropic chemical shift (δiso) values are calculated with respect to theoretical isotropic shielding constant (σiso) value of the tetramethylsilane (TMS). The average unsigned error in δiso values of the various oligomers varies between 2 and 5 ppm. The error of ca. 5.22 ppm for acrylonitrile arises mainly due to the cyano carbon. Oligomeric approach has been employed to calculate the δ iso values of the corresponding polymers. This approach is validated by the excellent correlation obtained for the linear fits. These calculated δiso values for the polymers are in good agreement with the experimental values.