Main-Factor Allocation and Coronal Realizability for Generalized Cospectral Mates of Trees
Abstract
We study how irreducible factors associated with main eigenvalues constrain the connected components of generalized cospectral mates. Let MG(x) be the main polynomial of a graph G, and let κm(G) denote the sum of the multiplicities in ϕG(x) of the irreducible factors dividing MG(x). We prove that every graph H with the same characteristic polynomial and main polynomial as G satisfies c(H)≤ κm(G). Consequently, if T is a tree and H is generalized cospectral with T, then β(H)=c(H)−1≤ κm(T)−1. We develop the case κm(T)=2 in detail. Any disconnected generalized cospectral mate is the union of a tree and a connected bipartite unicyclic graph, and the coronals of its two components are uniquely prescribed by the canonical decomposition of the coronal of T with respect to the two irreducible main factors. This turns the existence of a disconnected mate into a component-realizability problem. Factor moments yield realizability and tree-forcing obstructions, while matching data provide complementary characteristic-polynomial information. In particular, the unique cycle of any disconnected mate has length at least $6$, and finitely many matching identities, together with component-coronal realizability, certify generalized cospectrality under an explicit degree bound. As an application, we show that the double star D(2m,m+1) is DGS but not DS whenever m≥2 and neither m nor 2m+2 is a perfect square. In particular, D(8t+4,4t+3), t≥0, gives an explicit infinite family of such graphs.
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BibTeX
@misc{zhu2026mainfactor,
title = {{Main-Factor Allocation and Coronal Realizability for Generalized Cospectral Mates of Trees}},
author = {Chaochao Zhu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.24137},
eprint = {2608.24137},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.24137},
}
RIS
TY - UNPB
TI - Main-Factor Allocation and Coronal Realizability for Generalized Cospectral Mates of Trees
AU - Chaochao Zhu
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2608.24137
UR - https://arxiv.org/abs/2608.24137
AB - We study how irreducible factors associated with main eigenvalues constrain the connected components of generalized cospectral mates. Let M_G(x) be the main polynomial of a graph G, and let κ_m(G) denote the sum of the multiplicities in ϕ_G(x) of the irreducible factors dividing M_G(x). We prove that every graph H with the same characteristic polynomial and main polynomial as G satisfies c(H)≤ κ_m(G). Consequently, if T is a tree and H is generalized cospectral with T, then β(H)=c(H)−1≤ κ_m(T)−1. We develop the case κ_m(T)=2 in detail. Any disconnected generalized cospectral mate is the union of a tree and a connected bipartite unicyclic graph, and the coronals of its two components are uniquely prescribed by the canonical decomposition of the coronal of T with respect to the two irreducible main factors. This turns the existence of a disconnected mate into a component-realizability problem. Factor moments yield realizability and tree-forcing obstructions, while matching data provide complementary characteristic-polynomial information. In particular, the unique cycle of any disconnected mate has length at least $6$, and finitely many matching identities, together with component-coronal realizability, certify generalized cospectrality under an explicit degree bound. As an application, we show that the double star D(2m,m+1) is DGS but not DS whenever m≥2 and neither m nor 2m+2 is a perfect square. In particular, D(8t+4,4t+3), t≥0, gives an explicit infinite family of such graphs.
ER -