Tree Cover Number and Maximum Semidefinite Nullity of Some Graph Classes

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Let $G$ be a graph with a vertex set $V$ and an edge set $E$ consisting of unordered pairs of vertices. The tree cover number of $G$, denoted $\tau(G)$, is the minimum number of vertex disjoint simple trees occurring as induced subgraphs of $G$ that cover all the vertices of $G$. In this paper, the tree cover number of a line graph $\tau(L(G))$ is shown to be equal to the path number $\pi(G)$ of $G$. Also, the tree cover numbers of shadow graphs, corona and Cartesian product of two graphs are found.

The graph parameter $\tau(G)$ is related to another graph parameter $M_+(G)$, called the maximum semidefinite nullity of $G$. Suppose $S_+(G,\mathbb{R})$ denotes the collection of positive semidefinite real symmetric matrices associated with a given graph $G$. Then $M_+(G)$ is the maximum nullity among all matrices in $S_+(G,\mathbb{R})$. It has been conjectured that $\tau(G)\leq M_+(G)$. The conjecture is shown to be true for graph classes considered in this work.

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