Projective Dimension of Some Graphs

Thankachan, Reji , Rosemary, Ruby and Balakrishnan, Sneha

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creative_2023_32_1_87_96

DOI: https://doi.org/10.37193/CMI.2023.01.09

Issue no: Vol 32/2023 no. 1 Tags: Projective dimension, Edge ideals, Star, product graphs

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In this paper exact values for the projective dimension of edge ideals associated to some star related graphs and product graphs $G\ \square\ P_2$ , when $G=\ C_n,\ K_n$ and upper bounds for the projective dimension when $G=\ P_n,\ W_n$ , are obtained. We have proved that $pd(C_{n+1}\ \square\ P_2)= 2\big(n-\left\lfloor \frac{n}{4}\right\rfloor\big)$ , $pd(K_n\ \square\ P_2)= 2n-2$ and $pd(P_{n+1}\ \square \ P_2)\le n+3+\left\lfloor \frac{n-3}{2}\right\rfloor$ , $pd(W_n\ \square\ P_2)\leq n+1+\lceil\frac{2n-1}{3}\rceil$ . These values are functions of the number of vertices in the corresponding graphs.