A constructive method to determine the total vertex irregularity strength of two flower graph variants

Abstract

Graph labeling is a well-established area of research in discrete mathematics. One notable variant is total vertex irregular labeling, which assigns labels to both vertices and edges such that the resulting vertex weights are all distinct. In this paper, we propose a constructive method to determine the total vertex irregularity strength of two flower graph variants, each characterized by only two distinct vertex degrees. Our approach involves assigning explicit labels and verifying that the resulting weighted degrees differ at every vertex. This method not only determines the exact total vertex irregularity strength for each graph but also introduces a replicable labeling strategy applicable to other graphs with similar structural features. The main results in this paper are as follows:•<i>The total vertex irregularity strength of the modified sunflower graph is</i> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:mfrac><mml:mi>n</mml:mi> <mml:mn>2</mml:mn></mml:mfrac> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn></mml:mrow> </mml:math> <i>;</i>•<i>The total vertex irregularity strength of the flower petal graph is</i> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>⌈</mml:mo> <mml:mfrac> <mml:mrow> <mml:msup><mml:mrow><mml:mi>n</mml:mi></mml:mrow> <mml:mn>2</mml:mn></mml:msup> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn></mml:mrow> <mml:mn>3</mml:mn></mml:mfrac> <mml:mo>⌉</mml:mo></mml:mrow> </mml:math> <i>;</i>•<i>These values differ due to the fundamentally distinct structural properties of the to graphs.</i>The proposed method contributes to the broader study of irregular graph labeling and offers potential applications in network analysis and graph-based modeling.

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