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Volume 34, Issue 4
On the Normalized Laplacian Spectrum of a New Join of Two Graphs

Xianzhang Wu & Lili Shen

Ann. Appl. Math., 34 (2018), pp. 407-415.

Published online: 2022-06

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  • Abstract

Given graphs $G_1$ and $G_2,$ we define a graph operation on $G_1$ and $G_2$, namely the $SSG$-vertex join of $G_1$ and $G_2,$ denoted by $G_1 \star G_2.$ Let $S(G)$ be the subdivision graph of $G.$ The $SSG$-vertex join $G_1\star G_2$ is the graph obtained from $S(G_1)$ and $S(G_2)$ by joining each vertex of $G_1$ with each vertex of $G_2.$ In this paper, when $G_i (i = 1, 2)$ is a regular graph, we determine the normalized Laplacian spectrum of $G_1 \star G_2.$ As applications, we construct many pairs of normalized Laplacian cospectral graphs, the normalized Laplacian energy, and the degree Kirchhoff index of $G_1 \star G_2.$

  • Keywords

spectrum, $SSG$-vertex join, normalized Laplacian cospectral graphs, normalized Laplacian energy, degree Kirchhoff index.

  • AMS Subject Headings

05C07

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAM-34-407, author = {Wu , Xianzhang and Shen , Lili}, title = {On the Normalized Laplacian Spectrum of a New Join of Two Graphs}, journal = {Annals of Applied Mathematics}, year = {2022}, volume = {34}, number = {4}, pages = {407--415}, abstract = {

Given graphs $G_1$ and $G_2,$ we define a graph operation on $G_1$ and $G_2$, namely the $SSG$-vertex join of $G_1$ and $G_2,$ denoted by $G_1 \star G_2.$ Let $S(G)$ be the subdivision graph of $G.$ The $SSG$-vertex join $G_1\star G_2$ is the graph obtained from $S(G_1)$ and $S(G_2)$ by joining each vertex of $G_1$ with each vertex of $G_2.$ In this paper, when $G_i (i = 1, 2)$ is a regular graph, we determine the normalized Laplacian spectrum of $G_1 \star G_2.$ As applications, we construct many pairs of normalized Laplacian cospectral graphs, the normalized Laplacian energy, and the degree Kirchhoff index of $G_1 \star G_2.$

}, issn = {}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/aam/20588.html} }
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 -

Given graphs $G_1$ and $G_2,$ we define a graph operation on $G_1$ and $G_2$, namely the $SSG$-vertex join of $G_1$ and $G_2,$ denoted by $G_1 \star G_2.$ Let $S(G)$ be the subdivision graph of $G.$ The $SSG$-vertex join $G_1\star G_2$ is the graph obtained from $S(G_1)$ and $S(G_2)$ by joining each vertex of $G_1$ with each vertex of $G_2.$ In this paper, when $G_i (i = 1, 2)$ is a regular graph, we determine the normalized Laplacian spectrum of $G_1 \star G_2.$ As applications, we construct many pairs of normalized Laplacian cospectral graphs, the normalized Laplacian energy, and the degree Kirchhoff index of $G_1 \star G_2.$

Xianzhang Wu & Lili Shen. (2022). On the Normalized Laplacian Spectrum of a New Join of Two Graphs. Annals of Applied Mathematics. 34 (4). 407-415. doi:
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