On the maximal index of C4--free unbalanced signed graphs of prescribed size
Abstract
The extremal problem of determining the graphs with maximum spectral radius among all graphs with a prescribed number of edges was resolved by Rowlinson [Linear Algebra Appl. 110 (1988) 43--53]. A signed graph Γ=(G,σ) is called Ck--free if it contains no negative k-cycle as a signed subgraph. The index of a signed graph is defined as the largest eigenvalue of its adjacency matrix. This paper addresses the maximization of the index of unbalanced signed graphs. For signed graphs of a prescribed order, Wang and Lin [Discrete Appl. Math. 372 (2025) 164--172] determined the maximum index of C4−-free unbalanced signed graphs. Subsequently, Wang, Peng and Hou [Discrete Math. 349 (2026) 114871] characterized the unbalanced signed graphs of prescribed size with maximum index. This paper treats the edge spectral Turán-type problem for C4--free unbalanced signed graphs. The all-positive signed graph with underlying graph G is denoted by (G,+). Evidently, A((G,+))=A(G). In particular, (Hd,t,+) is the all-positive signed graph whose underlying graph is Hd,t, where Hd,0=Kd and for t>0, Hd,t is obtained from Kd by adding a single vertex adjacent to exactly t vertices of Kd. Let m−3=binomd2+t with 0≤ t≤ d−1. Up to switching isomorphism and the addition of isolated vertices, the unique extremal unbalanced signed graph is obtained by attaching a negative triangle to a specified vertex of the all-positive signed graph (Hd,t,+).
Related
- Maximum index of unbalanced signed graphs without a positive odd cycle
- Turán-type extremal problems for unbalanced signed graphs
- Spectral and edge conditions for negative Hamiltonian cycles in unbalanced signed graphs
- Unbalanced spectral Turán problem for color-critical graphs with prescribed large maximum degree
- Unbalanced Turán and spectral Turán problems with prescribed large maximum degree
BibTeX
@misc{yang2026maximal,
title = {{On the maximal index of $C_4^-$-free unbalanced signed graphs of prescribed size}},
author = {Zhihui Yang and Dan Li and Yuanyuan Chen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30881},
eprint = {2610.30881},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30881},
}
RIS
TY - UNPB
TI - On the maximal index of C_4^--free unbalanced signed graphs of prescribed size
AU - Zhihui Yang
AU - Dan Li
AU - Yuanyuan Chen
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.30881
UR - https://arxiv.org/abs/2610.30881
AB - The extremal problem of determining the graphs with maximum spectral radius among all graphs with a prescribed number of edges was resolved by Rowlinson [Linear Algebra Appl. 110 (1988) 43--53]. A signed graph Γ=(G,σ) is called C_k^--free if it contains no negative k-cycle as a signed subgraph. The index of a signed graph is defined as the largest eigenvalue of its adjacency matrix. This paper addresses the maximization of the index of unbalanced signed graphs. For signed graphs of a prescribed order, Wang and Lin [Discrete Appl. Math. 372 (2025) 164--172] determined the maximum index of C_4^−-free unbalanced signed graphs. Subsequently, Wang, Peng and Hou [Discrete Math. 349 (2026) 114871] characterized the unbalanced signed graphs of prescribed size with maximum index. This paper treats the edge spectral Turán-type problem for C_4^--free unbalanced signed graphs. The all-positive signed graph with underlying graph G is denoted by (G,+). Evidently, A((G,+))=A(G). In particular, (H_d,t,+) is the all-positive signed graph whose underlying graph is H_d,t, where H_d,0=K_d and for t>0, H_d,t is obtained from K_d by adding a single vertex adjacent to exactly t vertices of K_d. Let m−3=binomd2+t with 0≤ t≤ d−1. Up to switching isomorphism and the addition of isolated vertices, the unique extremal unbalanced signed graph is obtained by attaching a negative triangle to a specified vertex of the all-positive signed graph (H_d,t,+).
ER -