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Volume 58, Issue 3
A Note on the Determinant of a Special Class of $Q$-Walk Matrices

Guixian Tian, Junxing Wu, Shuyu Cui & Huilu Sun

DOI: 10.4208/jms.v58n3.25.02

J. Math. Study, 58 (2025), pp. 275-285.

Published online: 2025-09

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  • Abstract

For a graph $G$ of order $n,$ its $Q$-walk matrix is defined by $W_Q(G) = [e,Qe,···,Q^{n−1}e],$ where $Q$ is the signless Laplacian matrix of $G$ and $e$ denotes the all-one column vector. Let $G \circ P_k$ represent the rooted product graph of $G$ and a path $P_k.$ In this note, we establish the relationship between determinants of $W_Q(G)$ and $W_Q(G \circ P_k )$ for $k=2,3.$

  • Keywords

Signless Laplacian matrix, $Q$-walk matrix, rooted product graph, determinant.

  • AMS Subject Headings

05C50, 15A18

  • Copyright

COPYRIGHT: © Global Science Press

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  • TXT
@Article{JMS-58-275, author = {Tian , GuixianWu , JunxingCui , Shuyu and Sun , Huilu}, title = {A Note on the Determinant of a Special Class of $Q$-Walk Matrices}, journal = {Journal of Mathematical Study}, year = {2025}, volume = {58}, number = {3}, pages = {275--285}, abstract = {

For a graph $G$ of order $n,$ its $Q$-walk matrix is defined by $W_Q(G) = [e,Qe,···,Q^{n−1}e],$ where $Q$ is the signless Laplacian matrix of $G$ and $e$ denotes the all-one column vector. Let $G \circ P_k$ represent the rooted product graph of $G$ and a path $P_k.$ In this note, we establish the relationship between determinants of $W_Q(G)$ and $W_Q(G \circ P_k )$ for $k=2,3.$

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v58n3.25.02}, url = {http://global-sci.org/intro/article_detail/jms/24408.html} }
TY - JOUR T1 - A Note on the Determinant of a Special Class of $Q$-Walk Matrices AU - Tian , Guixian AU - Wu , Junxing AU - Cui , Shuyu AU - Sun , Huilu JO - Journal of Mathematical Study VL - 3 SP - 275 EP - 285 PY - 2025 DA - 2025/09 SN - 58 DO - http://doi.org/10.4208/jms.v58n3.25.02 UR - https://global-sci.org/intro/article_detail/jms/24408.html KW - Signless Laplacian matrix, $Q$-walk matrix, rooted product graph, determinant. AB -

For a graph $G$ of order $n,$ its $Q$-walk matrix is defined by $W_Q(G) = [e,Qe,···,Q^{n−1}e],$ where $Q$ is the signless Laplacian matrix of $G$ and $e$ denotes the all-one column vector. Let $G \circ P_k$ represent the rooted product graph of $G$ and a path $P_k.$ In this note, we establish the relationship between determinants of $W_Q(G)$ and $W_Q(G \circ P_k )$ for $k=2,3.$

Tian , GuixianWu , JunxingCui , Shuyu and Sun , Huilu. (2025). A Note on the Determinant of a Special Class of $Q$-Walk Matrices. Journal of Mathematical Study. 58 (3). 275-285. doi:10.4208/jms.v58n3.25.02
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