TY - JOUR
T1 - On the Normalized Laplacian Spectrum of a New Join of Two Graphs
AU - Wu , Xianzhang
AU - Shen , Lili
JO - Annals of Applied Mathematics
VL - 4
SP - 407
EP - 415
PY - 2022
DA - 2022/06
SN - 34
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/aam/20588.html
KW - spectrum, $SSG$-vertex join, normalized Laplacian cospectral
graphs, normalized Laplacian energy, degree Kirchhoff index.
AB - <p style="text-align: justify;">Given graphs $G_1$ and $G_2,$ we define a graph operation on <span style="text-align: justify;">$G_1$</span>&nbsp;and <span style="text-align: justify;">$G_2$</span>,
namely the $SSG$-vertex join of <span style="text-align: justify;">$G_1$</span>&nbsp;and <span style="text-align: justify;">$G_2,$</span>&nbsp;denoted by $G_1 \star G_2.$&nbsp;Let $S(G)$ be
the subdivision graph of $G.$ The $SSG$-vertex join $G_1\star G_2$&nbsp;is the graph obtained
from $S(G_1)$ and $S(G_2)$ by joining each vertex of <span style="text-align: justify;">$G_1$</span> with each vertex of <span style="text-align: justify;">$G_2.$</span> In
this paper, when $G_i (i = 1, 2)$ is a regular graph, we determine the normalized
Laplacian spectrum of $G_1 \star G_2.$&nbsp;As applications, we construct many pairs of
normalized Laplacian cospectral graphs, the normalized Laplacian energy, and
the degree Kirchhoff index of $G_1 \star G_2.$</p>