Vestnik КRAUNC. Fiz.-Mat. Nauki. 2026. vol. 54. no. 1. P. 33 – 43. ISSN 2079-6641

MATHEMATICS
https://doi.org/10.26117/2079-6641-2026-54-1-33-43
Research Article
Full text in English
MSC 05C07

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The Neighbor Degree Sum — Distance Index of the Graph

T. V. Asha^{\ast}¹, B. Chaluvaraju²

¹Government First Grade College, K. R. Pura, Bangalore, 560036, India
²Bangalore University, Jnana Bharathi Campus, Bangalore, 560056, India

Abstract. In this paper, we introduce three novel topological indices for a connected undirected graph G without loops or multiple edges. The indices are based on the sum of neighbor degrees and the distances between vertices, offering an enhancement over classical degree-based graph invariants. For each vertex, the neighbor degree is defined as the sum of the degrees of its adjacent vertices. Using this quantity together with pairwise distances, we define: the neighbor Schultz index NDD(G), which sums the neighbor degrees of each vertex pair multiplied by their distance; the neighbor Gutman index NZZ(G), where the sum of neighbor degrees is replaced by their product; and the neighbor Wiener–Albertson index NWA(G), which sums the absolute difference of neighbor degrees multiplied by the distance. Explicit formulas are derived for several standard graph families: complete graphs, complete bipartite graphs, stars, and wheels. The neighbor Wiener–Albertson index vanishes for regular graphs, making it a natural measure of graph irregularity. Sharp bounds for the new indices are established in terms of the number of vertices, diameter, minimum and maximum degrees, and the corresponding neighbor-degree parameters. It is shown, for instance, that the values of these indices for any connected graph lie between bounds expressed solely through the number of vertices and the extreme neighbor degrees. We conclude by outlining future research directions, including the application of these indices to chemical graphs, cactus graphs, cactus chains, Husimi trees, and their behavior in domination-related problems.

Key words: neighbor Schultz index, neighbor Gutman index, neighbor Wiener-Albertson index.

Received: 01.01.2026; Revised: 23.01.2026; Accepted: 29.01.2026; First online: 29.03.2026

For citation. Asha T. V. Chaluvaraju B. The neighbor degree sum — distance index of the graph. Vestnik KRAUNC. Fiz.-mat. nauki. 2026, 54: 1, 33-43. EDN: TTTUMU. https://doi.org/10.26117/2079-6641-2026-54-1-33-43.

Funding. Not funding

Competing interests. There are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. All authors contributed to this article. Authors are solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by all authors.

^{\ast}Correspondence: E-mail: ashagowda0403@gmail.com

The content is published under the terms of the Creative Commons Attribution 4.0 International License

© Asha T. V. Chaluvaraju B., 2026

© Institute of Cosmophysical Research and Radio Wave Propagation, 2026 (original layout, design, compilation)

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