A signed graph is a graph whose edges are labeled by signs. This is a bibliography of signed graphs and related mathematics. Several kinds of labelled graph have been called "signed" yet are mathematically very different. I distinguish four types: Group-signed graphs: the edge labels are elements of a 2-element group and are multiplied around a polygon (or along any walk). Among the natural generalizations are larger groups and vertex signs. Sign-colored graphs, in which the edges are labelled from a two-element set that is acted upon by the sign group: - interchanges labels, + leaves them unchanged. This is the kind of "signed graph" found in knot theory. The natural generalization is to more colors and more general groups —  or no group. Weighted graphs, in which the edge labels are the elements +1 and -1 of the integers or another additive domain. Weights behave like numbers, not signs; thus I regard work on weighted graphs as outside the scope of the bibliography — ex cept (to some extent) when the author calls the weights "signs". Labelled graphs where the labels have no structure or properties but are called "signs" for any or no reason.

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A Mathematical Bibliography of Signed and Gain Graphs and Allied Areas

Semi-Inner Products, that can be naturally defined in general Banach spaces over the real or complex number field, play an important role in describing the geometric properties of these spaces.In the first chapter of the book, a short introduction to the main properties of the duality mapping that will be used in the next chapters is given. Chapter 2 is devoted to the semi-inner products in the sense of Lumer-Giles while the 3rd chapter is concerning with the main properties of the superior and inferior semi-inner products. In the next chapter the main properties of Milicics semi-inner product and the properties of normed spaces of () — type are presented. The next two chapters investigate the geometric properties of (), ()and 2–inner product spaces introduced by the author, while Chapter 7 is entirely devoted to the study of different mappings that can naturally be associated to the norm derivatives in general normed spaces and, in particular, in inner product spaces. Chapters 8 and 9 investigate different orthogonalities that may be introduced in normed spaces and their intimate relationship with semi-inner products. In Chapter 11, orthogonal decomposition theorems in general normed spaces are provided, while in the next chapter the problem of approximating continuous linear functionals in general normed spaces and characterizations of reflexivity in this context are given. A deeper insight on this problem is then considered in Chapter 13, where some classes of continuous functionals are introduced and a density result based on the famous Bishop-Phelps theorem is obtained. In Chapter 14, the class of smooth normed spaces of (BD)-type and their application for non-linear operators is presented. In the next chapter the continuous sublinear functionals defined in Reflexive Banach spaces is investigated, while Chapter 16 deals with convex functions defined in more general spaces endowed with subinner products. The monograph concludes by considering the representation problem of linear forms defined on modules endowed with general semi-subinner products.

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Semi-Inner Products and Applications

Biological and medical imaging by nmr

Beyond the Zagreb indices

Inspired by recent work of augmented Zagreb index (AZI) we propose here a new topological index, Sanskruti index $$\mathcal {S}(G)$$
 of a molecular graph G. In QSAR/QSPR study, topological indices are utilized to guess the bioactivity of chemical compounds. The Sanskruti index $$\mathcal {S}(G)$$
 shows good correlation with entropy of an octane isomers. In this paper we compute the Sanskruti index $$\mathcal {S}(G)$$
 of line graphs of subdivison graphs of 2D-lattice, nanotube and nanotorus of $$\textit{TUC}_{4}C_{8}[p,q]$$
 .

Computing Sanskruti index of certain nanostructures

On the Zagreb indices of the line graphs of the subdivision graphs

Recently, nano structures have opened new dimensions in industry, electronics, and pharmaceutical and biological therapeutics The topological graph indices are numerical tendencies that often depict quantitative structural activity/property/toxicity relationships and correlate certain physio-chemical properties such as boiling point, stability, and strain energy, of respective nano-material In this article, we established closed forms of various degree-based topological graph indices of nano-structures of 2D-lattice, nano-tube and nano-torus of TUC4C8[r,s]

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Operations of Nanostructures via SDD, ABC4 and GA5 indices

In general, the relations among Zagreb polynomials on three graph operators are discussed in this paper. Specifically, relations between Zagreb polynomials of a graph $G$ and a graph obtained by applying the operators $S(G)$ , $R(G)$ and $Q(G)$ are investigated. In a separate section, the relation between Zagreb polynomial of a graph $G$ and its corona is also described.

/pdf/zagreb-polynomials-of-three-graph-operators-5bsvrqe7k7.pdf

Zagreb Polynomials of Three Graph Operators

‎‎The harmonic index of a connected graph $G$‎, ‎denoted by $H(G)$‎, ‎is‎ ‎defined as $H(G)=sum_{uvin E(G)}frac{2}{d_u+d_v}$‎ ‎where $d_v$ is the degree of a vertex $v$ in G‎. ‎In this paper‎, ‎expressions for the Harary indices of the‎ ‎join‎, ‎corona product‎, ‎Cartesian product‎, ‎composition and symmetric difference of graphs are‎ ‎derived‎.

https://toc.ui.ac.ir/article_7389_3e9ec295be34a42ee71a0570cc2fbfa9.pdf

On the harmonic index of graph operations

The aim of this paper was to derive new identities and relations associated with the q-Bernstein polynomials, q-Frobenius–Euler polynomials, l-functions, and q-Stirling numbers of the second kind. We also give some applications related to theses polynomials and numbers. Copyright © 2012 John Wiley & Sons, Ltd.

V. Lokesha

Papers

On the Zagreb indices of the line graphs of the subdivision graphs

Operations of Nanostructures via SDD, ABC4 and GA5 indices

Zagreb Polynomials of Three Graph Operators

On the harmonic index of graph operations

q-Bernstein polynomials related to q-Frobenius–Euler polynomials, l-functions, and q-Stirling numbers