Bond incident degree indices of stepwise irregular graphs

Damchaa Adiyanyam; Enkhbayar Azjargal; Lkhagva Buyantogtokh; Damchaa Adiyanyam; Enkhbayar Azjargal; Lkhagva Buyantogtokh

doi:10.3934/math.2022485

AIMS Mathematics

2022, Volume 7, Issue 5: 8685-8700. doi: 10.3934/math.2022485

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Research article

Bond incident degree indices of stepwise irregular graphs

1.
Department of Mathematics and Natural Sciences, Mongolian National University of Education, Baga toiruu-14, Ulaanbaatar, Mongolia
2.
Department of Mathematics, Mongolian National University of Education, Baga toiruu-14, Ulaanbaatar, Mongolia

Received: 06 November 2021 Revised: 13 January 2022 Accepted: 16 January 2022 Published: 02 March 2022
MSC : 05C07, 05C09, 05C35, 05C75

The bond incident degree (BID) index of a graph $ G $ is defined as $ BID_{f}(G) = \sum_{uv\in E(G)}f(d(u), d(v)) $, where $ d(u) $ is the degree of a vertex $ u $ and $ f $ is a non-negative real valued symmetric function of two variables. A graph is stepwise irregular if the degrees of any two of its adjacent vertices differ by exactly one. In this paper, we give a sharp upper bound on the maximum degree of stepwise irregular graphs of order $ n $ when $ n\equiv 2({\rm{mod}}\;4) $, and we give upper bounds on $ BID_{f} $ index in terms of the order $ n $ and the maximum degree $ \Delta $. Moreover, we completely characterize the extremal stepwise irregular graphs of order $ n $ with respect to $ BID_{f} $.
- maximum degree,
- stepwise irregular graph,
- BID index,
- bipartite graph
Citation: Damchaa Adiyanyam, Enkhbayar Azjargal, Lkhagva Buyantogtokh. Bond incident degree indices of stepwise irregular graphs[J]. AIMS Mathematics, 2022, 7(5): 8685-8700. doi: 10.3934/math.2022485

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Abstract

The bond incident degree (BID) index of a graph $ G $ is defined as $ BID_{f}(G) = \sum_{uv\in E(G)}f(d(u), d(v)) $, where $ d(u) $ is the degree of a vertex $ u $ and $ f $ is a non-negative real valued symmetric function of two variables. A graph is stepwise irregular if the degrees of any two of its adjacent vertices differ by exactly one. In this paper, we give a sharp upper bound on the maximum degree of stepwise irregular graphs of order $ n $ when $ n\equiv 2({\rm{mod}}\;4) $, and we give upper bounds on $ BID_{f} $ index in terms of the order $ n $ and the maximum degree $ \Delta $. Moreover, we completely characterize the extremal stepwise irregular graphs of order $ n $ with respect to $ BID_{f} $.

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