On the general sum-connectivity index of hypergraphs

Hongzhuan Wang; Piaoyang Yin; Yan Li; Hongzhuan Wang; Piaoyang Yin; Yan Li

doi:10.3934/math.2025983

AIMS Mathematics

2025, Volume 10, Issue 9: 22092-22105. doi: 10.3934/math.2025983

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Research article

On the general sum-connectivity index of hypergraphs

1.
Faculty of Mathematics and Physics, Huaiyin Institute of Technology, Huai'an 223003, Jiangsu, China
2.
School of Business, Huaiyin Institute of Technology, Huai'an 223003, Jiangsu, China

Received: 30 July 2025 Revised: 10 September 2025 Accepted: 16 September 2025 Published: 23 September 2025
MSC : 05C35, 05C50, 05C90

Given a non-zero real number $ \alpha $, the general sum-connectivity index $ \chi_{\alpha} $ for graph $ G $ is given by the sum $ \Sigma_{xy\in {E(G)}} (d_x+d_y)^{\alpha} $. Here, $ d_x $ denotes the degree of a vertex $ x $ in graph $ G $, and $ E(G) $ is the edge set of $ G $. A hypertree is a connected hypergraph without cycles. In a $ k $-uniform hypergraph, every hyperedge contains exactly $ k $ vertices, and a hypergraph is termed linear if any two distinct hyperedges share at most one vertex. This study addresses analytical challenges inherent in hypergraphs by concentrating on key subclasses and introducing innovative perturbation methods to solve fundamental extremal problems within these frameworks. Through this approach, we aim to broaden the scope and utility of graph-theoretic techniques. Specifically, within the class of uniform linear hypergraphs, we characterize the extremal hypergraphs with respect to the $ \chi_{\alpha} $ operation. Furthermore, we investigate extremal problems related to $ \chi_{\alpha} $ in both general hypergraphs and bipartite hypergraphs, yielding new insights into their previously unexplored structural properties.
- $ k $-uniform hypergraph,
- bipartite hypergraphs,
- general sum-connectivity index,
- extremal problem
Citation: Hongzhuan Wang, Piaoyang Yin, Yan Li. On the general sum-connectivity index of hypergraphs[J]. AIMS Mathematics, 2025, 10(9): 22092-22105. doi: 10.3934/math.2025983

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Abstract

Given a non-zero real number $ \alpha $, the general sum-connectivity index $ \chi_{\alpha} $ for graph $ G $ is given by the sum $ \Sigma_{xy\in {E(G)}} (d_x+d_y)^{\alpha} $. Here, $ d_x $ denotes the degree of a vertex $ x $ in graph $ G $, and $ E(G) $ is the edge set of $ G $. A hypertree is a connected hypergraph without cycles. In a $ k $-uniform hypergraph, every hyperedge contains exactly $ k $ vertices, and a hypergraph is termed linear if any two distinct hyperedges share at most one vertex. This study addresses analytical challenges inherent in hypergraphs by concentrating on key subclasses and introducing innovative perturbation methods to solve fundamental extremal problems within these frameworks. Through this approach, we aim to broaden the scope and utility of graph-theoretic techniques. Specifically, within the class of uniform linear hypergraphs, we characterize the extremal hypergraphs with respect to the $ \chi_{\alpha} $ operation. Furthermore, we investigate extremal problems related to $ \chi_{\alpha} $ in both general hypergraphs and bipartite hypergraphs, yielding new insights into their previously unexplored structural properties.

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