1Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, Iran 2Department of School Education, JK Govt. Kashmir, India 3Department of Mathematics, University of Kashmir, Srinagar, Kashmir, India
Acta mathematica Universitatis Comenianae, Tome 92 (2023) no. 3, pp. 197-211
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Maryam Baghipur; Modjtaba Ghorbani; Hilal Ganie; Shariefuddin Pirzada; Maryam Baghipur; Modjtaba Ghorbani; Hilal Ganie; Shariefuddin Pirzada. On the spread of the generalized adjacency matrix of a graph. Acta mathematica Universitatis Comenianae, Tome 92 (2023) no. 3, pp. 197-211. http://geodesic.mathdoc.fr/item/AMUC_2023_92_3_a0/
@article{AMUC_2023_92_3_a0,
author = {Maryam Baghipur and Modjtaba Ghorbani and Hilal Ganie and Shariefuddin Pirzada and Maryam Baghipur and Modjtaba Ghorbani and Hilal Ganie and Shariefuddin Pirzada},
title = { On the spread of the generalized adjacency matrix of a graph},
journal = {Acta mathematica Universitatis Comenianae},
pages = {197--211},
year = {2023},
volume = {92},
number = {3},
url = {http://geodesic.mathdoc.fr/item/AMUC_2023_92_3_a0/}
}
TY - JOUR
AU - Maryam Baghipur
AU - Modjtaba Ghorbani
AU - Hilal Ganie
AU - Shariefuddin Pirzada
AU - Maryam Baghipur
AU - Modjtaba Ghorbani
AU - Hilal Ganie
AU - Shariefuddin Pirzada
TI - On the spread of the generalized adjacency matrix of a graph
JO - Acta mathematica Universitatis Comenianae
PY - 2023
SP - 197
EP - 211
VL - 92
IS - 3
UR - http://geodesic.mathdoc.fr/item/AMUC_2023_92_3_a0/
ID - AMUC_2023_92_3_a0
ER -
%0 Journal Article
%A Maryam Baghipur
%A Modjtaba Ghorbani
%A Hilal Ganie
%A Shariefuddin Pirzada
%A Maryam Baghipur
%A Modjtaba Ghorbani
%A Hilal Ganie
%A Shariefuddin Pirzada
%T On the spread of the generalized adjacency matrix of a graph
%J Acta mathematica Universitatis Comenianae
%D 2023
%P 197-211
%V 92
%N 3
%U http://geodesic.mathdoc.fr/item/AMUC_2023_92_3_a0/
%F AMUC_2023_92_3_a0
Let $\mathsf{D}(G)$ and $\mathsf{A}(G)$, respectively, be the diagonal matrix of vertex degrees and the adjacency matrix of a connected graph $G$. The generalized adjacency matrix of $G$ is defined as $\mathsf{A}_{\alpha}(G)=\alpha \mathsf{D}(G)+(1-\alpha)\mathsf{A}(G)$, $\alpha\in [0,1]$. The spread of the generalized adjacency matrix, denoted by $\mathsf{S}(\mathsf{A}_{\alpha}(G))$, is defined as the difference of the largest and smallest eigenvalues of $\mathsf{A}_{\alpha}(G)$. The spread $ \mathsf{S}(\mathsf{Q}(G))$ of $\mathsf{Q}(G)$, the signless Laplacian matrix of $G$ and $ \mathsf{S}(\mathsf{A}(G))$ of $\mathsf{A}(G)$ are similarly defined. In this paper, we establish the relationships between $\mathsf{S}(\mathsf{A}_{\alpha}(G))$, $\mathsf{S}(\mathsf{Q}(G))$ and $\mathsf{S}(\mathsf{A}(G))$. Further, we find bounds for $\mathsf{S}(\mathsf{A}_{\alpha}(G))$ involving different parameters of $G$. Also, we obtain lower bounds for $\mathsf{S}(\mathsf{A}_{\alpha}(G))$ in terms of the clique number and independence number of $G$, and we characterize the extremal graphs in some cases.
