Article

# General Properties on Differential Sets of a Graph

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Faculty of Mathematics, Autonomous University of Guerrero, Carlos E. Adame 5, Col. La Garita, Acapulco 39650, Mexico
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Department of Economy, Quantitative Methods and Economic History, Pablo de Olavide University, Carretera de Utrera Km. 1, 41013 Sevilla, Spain
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Author to whom correspondence should be addressed.
Axioms 2021, 10(4), 265; https://doi.org/10.3390/axioms10040265
Received: 4 August 2021 / Revised: 5 October 2021 / Accepted: 11 October 2021 / Published: 20 October 2021
Let $G=\left(V,E\right)$ be a graph, and let $\beta \in \mathbb{R}$. Motivated by a service coverage maximization problem with limited resources, we study the $\beta$-differential of G. The β-differential of G, denoted by ${\partial }_{\beta }\left(G\right)$, is defined as ${\partial }_{\beta }\left(G\right):=max\left\{|B\left(S\right)|-\beta |S|\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}suchthat\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}S\subseteq V\right\}$. The case in which $\beta =1$ is known as the differential of G, and hence ${\partial }_{\beta }\left(G\right)$ can be considered as a generalization of the differential $\partial \left(G\right)$ of G. In this paper, upper and lower bounds for ${\partial }_{\beta }\left(G\right)$ are given in terms of its order $|G|$, minimum degree $\delta \left(G\right)$, maximum degree $\Delta \left(G\right)$, among other invariants of G. Likewise, the $\beta$-differential for graphs with heavy vertices is studied, extending the set of applications that this concept can have. View Full-Text
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MDPI and ACS Style

Basilio, L.A.; Bermudo, S.; Hernández-Gómez, J.C.; Sigarreta, J.M. General Properties on Differential Sets of a Graph. Axioms 2021, 10, 265. https://doi.org/10.3390/axioms10040265

AMA Style

Basilio LA, Bermudo S, Hernández-Gómez JC, Sigarreta JM. General Properties on Differential Sets of a Graph. Axioms. 2021; 10(4):265. https://doi.org/10.3390/axioms10040265

Chicago/Turabian Style

Basilio, Ludwin A., Sergio Bermudo, Juan C. Hernández-Gómez, and José M. Sigarreta 2021. "General Properties on Differential Sets of a Graph" Axioms 10, no. 4: 265. https://doi.org/10.3390/axioms10040265

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