Maximizing general first Zagreb and sum-connectivity indices for unicyclic graphs with given independence number

Let $K$ be any finite field. For any prime $p$, the $p$-adic valuation map is given by $\psi_{p}:K/\{0\} \to \R^+\bigcup\{0\}$ is given by $\psi_{p}(r) = n$ where $r = p^n \frac{a}{b}$, where $p,a,b$ are relatively prime. The field $K$ together with a valuation is called valued field. Also, any field $K$ has the trivial valuation determined by $\psi{(K)} = \{0,1\}$. Through out the paper K represents $\Z_q$. In this paper, we construct the graph corresponding to the valuation map called the valued field graph, denoted by $VFG_{p}(\Z_{q})$ whose vertex set is $\{v_0,v_1,v_2,\ldots, v_{q-1}\}$ where two vertices $v_i$ and $v_j$ are adjacent if $\psi_{p}(i) = j$ or $\psi_{p}(j) = i$. Here, we tried to characterize the valued field graph in $\Z_q$. Also we analyse various graph theoretical parameters such as diameter, independence number etc.

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The Antifungal Activity of Some Aliphatic and Aromatic Acids

Collection of Czechoslovak Chemical Communications ◽

10.1135/cccc19921134 ◽

1992 ◽

Vol 57 (5) ◽

pp. 1134-1142 ◽

Cited By ~ 9

Author(s):

Bohuslav Rittich ◽

Marta Pirochtová ◽

Jiří Hřib ◽

Kamila Jurtíková ◽

Petr Doležal

Keyword(s):

Biological Activities ◽

Pure Water ◽

Water System ◽

Penicillium Expansum ◽

Aromatic Acids ◽

Antifungal Activities ◽

Aliphatic Acids ◽

Significant Relationships ◽

Physico Chemical ◽

Connectivity Indices

The present paper deals with the relationship between biological activities of some aliphatic and aromatic acids and their physico-chemical parameters expressing the influence of hydrophobic factors. The test strain in the biotest of growth inhibition was the fungus Fusarium moniliforme CCMF-180 and Penicillium expansum CCMF-576. Significant relationship between antifungal activities of un-ionized form of aliphatic acids and their capacity factors (log k'0) extrapolated to pure water, partition coefficients determined in 1-octanol-water system (log Poct) and the first order of molecular connectivity indices (1χ) were calculated. The ionized form of aliphatic acids were antifungally active too. For benzoic acids significant relationships between antifungal activities and capacity factors of anionic form (log k'ia) were calculated.

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Mathematical Properties of Variable Topological Indices

Symmetry ◽

10.3390/sym13010043 ◽

2020 ◽

Vol 13 (1) ◽

pp. 43

Author(s):

José M. Sigarreta

Keyword(s):

Real Number ◽

Large Class ◽

Symmetric Functions ◽

Topological Indices ◽

Current Interest ◽

Zagreb Indices ◽

Mathematical Properties ◽

Connectivity Indices ◽

Optimal Bounds ◽

New Framework

A topic of current interest in the study of topological indices is to find relations between some index and one or several relevant parameters and/or other indices. In this paper we study two general topological indices Aα and Bα, defined for each graph H=(V(H),E(H)) by Aα(H)=∑ij∈E(H)f(di,dj)α and Bα(H)=∑i∈V(H)h(di)α, where di denotes the degree of the vertex i and α is any real number. Many important topological indices can be obtained from Aα and Bα by choosing appropriate symmetric functions and values of α. This new framework provides new tools that allow to obtain in a unified way inequalities involving many different topological indices. In particular, we obtain new optimal bounds on the variable Zagreb indices, the variable sum-connectivity index, the variable geometric-arithmetic index and the variable inverse sum indeg index. Thus, our approach provides both new tools for the study of topological indices and new bounds for a large class of topological indices. We obtain several optimal bounds of Aα (respectively, Bα) involving Aβ (respectively, Bβ). Moreover, we provide several bounds of the variable geometric-arithmetic index in terms of the variable inverse sum indeg index, and two bounds of the variable inverse sum indeg index in terms of the variable second Zagreb and the variable sum-connectivity indices.

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Independence and domination on shogiboard graphs

Recreational Mathematics Magazine ◽

10.1515/rmm-2017-0018 ◽

2017 ◽

Vol 4 (8) ◽

pp. 25-37 ◽

Cited By ~ 1

Author(s):

Doug Chatham

Keyword(s):

Independence Number ◽

Domination Number ◽

The Other ◽

Independent Domination

Abstract Given a (symmetrically-moving) piece from a chesslike game, such as shogi, and an n×n board, we can form a graph with a vertex for each square and an edge between two vertices if the piece can move from one vertex to the other. We consider two pieces from shogi: the dragon king, which moves like a rook and king from chess, and the dragon horse, which moves like a bishop and rook from chess. We show that the independence number for the dragon kings graph equals the independence number for the queens graph. We show that the (independent) domination number of the dragon kings graph is n − 2 for 4 ≤ n ≤ 6 and n − 3 for n ≥ 7. For the dragon horses graph, we show that the independence number is 2n − 3 for n ≥ 5, the domination number is at most n−1 for n ≥ 4, and the independent domination number is at most n for n ≥ 5.

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