Neighbourhood second Zagreb matrix of graphs: spectral radius bounds and extremal graphs
Abstract
Let G be a connected graph on at least two vertices, and let δ(i) be the sum of the degrees of the neighbours of a vertex i. The \emph{neighbourhood second Zagreb matrix} NF(G) has (i,j)-entry δ(i)δ(j) if i∼ j and $0$ otherwise; its spectral radius is denoted by rNF(G). We prove that rNF(G)≥ρ(G)5, where ρ(G) is the adjacency spectral radius, with equality if and only if G is regular or bipartite semiregular, and we use this to show that the same graphs are exactly the extremal graphs of a Berman--Zhang--type upper bound. We sharpen the estimates demin2ρ(G)≤rNF(G)≤demax2ρ(G) by means of the extreme edge weights of NF(G), give sharp upper bounds in terms of the maximum degree dmax, the maximum neighbourhood degree sum demax, the size m and the clique number ω(G), and determine the extremal graphs for bipartite graphs and trees. For every connected unicyclic graph G of order n≥3 we prove 32=rNF(Cn)≤rNF(G)≤rNF(Sp n), where Sp n is the triangle with n−3 pendant edges at one vertex, with equality only for Cn and only for Sp n, respectively. Finally, we give bounds in terms of the first Zagreb index and for the neighbourhood second Zagreb energy; for lower energy bounds, the extremal graphs are described completely.
Related
- The Maximum Second Zagreb Spectral Radius of Trees, Unicyclic Graphs and Bicyclic Graphs
- New Inequalities Linking the First and Second Zagreb Indices
- Extremal Rota-Baxter operators on matrix algebras
- A general theorem in spectral extremal graph theory
- Exact Values, Extremal Classifications, and Sum-of-Squares Reductions for Second-Order Zarankiewicz Numbers
BibTeX
@misc{chakraborty2026neighbourhood,
title = {{Neighbourhood second Zagreb matrix of graphs: spectral radius bounds and extremal graphs}},
author = {Subhendu Chakraborty and Priti Prasanna Mondal},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06091},
eprint = {2611.06091},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06091},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Neighbourhood second Zagreb matrix of graphs: spectral radius bounds and extremal graphs
AU - Subhendu Chakraborty
AU - Priti Prasanna Mondal
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06091
UR - https://arxiv.org/abs/2611.06091
AB - Let G be a connected graph on at least two vertices, and let δ(i) be the sum of the degrees of the neighbours of a vertex i. The \emph{neighbourhood second Zagreb matrix} NF(G) has (i,j)-entry δ(i)δ(j) if i∼ j and $0$ otherwise; its spectral radius is denoted by rNF(G). We prove that rNF(G)≥ρ(G)^5, where ρ(G) is the adjacency spectral radius, with equality if and only if G is regular or bipartite semiregular, and we use this to show that the same graphs are exactly the extremal graphs of a Berman--Zhang--type upper bound. We sharpen the estimates demin^2ρ(G)≤rNF(G)≤demax^2ρ(G) by means of the extreme edge weights of NF(G), give sharp upper bounds in terms of the maximum degree dmax, the maximum neighbourhood degree sum demax, the size m and the clique number ω(G), and determine the extremal graphs for bipartite graphs and trees. For every connected unicyclic graph G of order n≥3 we prove 32=rNF(C_n)≤rNF(G)≤rNF(Sp n), where Sp n is the triangle with n−3 pendant edges at one vertex, with equality only for C_n and only for Sp n, respectively. Finally, we give bounds in terms of the first Zagreb index and for the neighbourhood second Zagreb energy; for lower energy bounds, the extremal graphs are described completely.
ER -