Reversing the Mostar line-graph inequality with long pendant paths
Abstract
Let Kt be obtained by attaching a pendant path of length t to a fixed rooted graph K. We prove that Mo(L(Kt))−Mo(Kt), the difference between its line-graph and original Mostar indices, is exactly affine on each parity class beyond a sharp uniform cutoff. The slope depends on distance-level neighbor counts and is positive for every non-bipartite core. Triangle chains give infinitely many graphs with Mo(L(G))>Mo(G) at every positive cyclomatic number c, with maximum degree three, resolving Alex-Indulal Problem 3.3. At zero slope, a cactus branch-mass formula decides equality; identical vertex profiles need not give identical intercepts. Fixed cores giving equality for all sufficiently long attachments exist exactly when c is odd. Maximum degree three suffices, and at every even c a binary-tree construction gives equality on one eventual parity class.
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BibTeX
@misc{kat2026reversing,
title = {{Reversing the Mostar line-graph inequality with long pendant paths}},
author = {Kerem Kat},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00829},
eprint = {2610.00829},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00829},
}
RIS
TY - UNPB
TI - Reversing the Mostar line-graph inequality with long pendant paths
AU - Kerem Kat
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00829
UR - https://arxiv.org/abs/2610.00829
AB - Let K_t be obtained by attaching a pendant path of length t to a fixed rooted graph K. We prove that Mo(L(K_t))−Mo(K_t), the difference between its line-graph and original Mostar indices, is exactly affine on each parity class beyond a sharp uniform cutoff. The slope depends on distance-level neighbor counts and is positive for every non-bipartite core. Triangle chains give infinitely many graphs with Mo(L(G))>Mo(G) at every positive cyclomatic number c, with maximum degree three, resolving Alex-Indulal Problem 3.3. At zero slope, a cactus branch-mass formula decides equality; identical vertex profiles need not give identical intercepts. Fixed cores giving equality for all sufficiently long attachments exist exactly when c is odd. Maximum degree three suffices, and at every even c a binary-tree construction gives equality on one eventual parity class.
ER -