The Maximum Second Zagreb Spectral Radius of Trees, Unicyclic Graphs and Bicyclic Graphs
Abstract
The second Zagreb matrix of a graph G is Z(G)=D(G)A(G)D(G), where D(G) and A(G) are the degree matrix and the adjacency matrix of G. For many degree-based weighted adjacency matrices, the trees, unicyclic graphs and bicyclic graphs with the largest spectral radius have a vertex adjacent to all others, and Mei et al. conjectured this for every such matrix on unicyclic graphs. Jiang et al. showed that Z(G) is a counterexample. We prove that, for q∈{0,1,2} and every n≥4, the connected graph of order n and cyclomatic number q maximizing the spectral radius of Z(G) is unique, and is obtained from two adjacent vertices with q common neighbours by attaching n−q−2 pendent vertices to them as evenly as possible. The only exceptions are the star for q=0 and n≤7, and the triangle with one pendent vertex at each vertex for q=1 and n=6. This confirms a conjecture of Jiang et al.\ and all three parts of a conjecture of Shen and Shan for this matrix, with best possible thresholds on n. For n≥8, the maximum value is ⌊ (n+q)2/4⌋+(n+q)/2+7q−2+O(1/n), whereas the star gives only (n−1)3/2. A key tool is a determinant criterion that reduces spectral comparisons to sign conditions on explicit polynomials.
Related
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BibTeX
@misc{wang2026maximum,
title = {{The Maximum Second Zagreb Spectral Radius of Trees, Unicyclic Graphs and Bicyclic Graphs}},
author = {Juan Wang and Genjie Wang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05057},
eprint = {2611.05057},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05057},
}
RIS
TY - UNPB
TI - The Maximum Second Zagreb Spectral Radius of Trees, Unicyclic Graphs and Bicyclic Graphs
AU - Juan Wang
AU - Genjie Wang
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05057
UR - https://arxiv.org/abs/2611.05057
AB - The second Zagreb matrix of a graph G is Z(G)=D(G)A(G)D(G), where D(G) and A(G) are the degree matrix and the adjacency matrix of G. For many degree-based weighted adjacency matrices, the trees, unicyclic graphs and bicyclic graphs with the largest spectral radius have a vertex adjacent to all others, and Mei et al. conjectured this for every such matrix on unicyclic graphs. Jiang et al. showed that Z(G) is a counterexample. We prove that, for q∈{0,1,2} and every n≥4, the connected graph of order n and cyclomatic number q maximizing the spectral radius of Z(G) is unique, and is obtained from two adjacent vertices with q common neighbours by attaching n−q−2 pendent vertices to them as evenly as possible. The only exceptions are the star for q=0 and n≤7, and the triangle with one pendent vertex at each vertex for q=1 and n=6. This confirms a conjecture of Jiang et al.\ and all three parts of a conjecture of Shen and Shan for this matrix, with best possible thresholds on n. For n≥8, the maximum value is ⌊ (n+q)^2/4⌋+(n+q)/2+7q−2+O(1/n), whereas the star gives only (n−1)^3/2. A key tool is a determinant criterion that reduces spectral comparisons to sign conditions on explicit polynomials.
ER -