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Given a connected regular graph $G$, let $l(G)$ be its line graph, $s(G)$ its subdivision graph, $r(G)$ the graph obtained from $G$ by adding a new vertex corresponding to each edge of $G$ and joining each new vertex to the end vertices of the corresponding edge and $q(G)$ the graph obtained from $G$ by inserting a new vertex into every edge of $G$ and new edges joining the pairs of new vertices which lie on adjacent edges of $G$. A formula for the normalised Laplacian characteristic polynomial of $l(G)$ (respectively $s(G),r(G)$ and $q(G)$) in terms of the normalised Laplacian characteristic polynomial of $G$ and the number of vertices and edges of $G$ is developed and used to give a sharp lower bound for the degree-Kirchhoff index and a formula for the number of spanning trees of $l(G)$ (respectively $s(G),r(G)$ and $q(G)$).

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Anoectochilus roxburghii (Wall.) Lindl. (Orchidaceae) is an endangered medicinal plant in China, also called “King Medicine”. Due to lacking of sufficient nutrients in dust-like seeds, orchid species depend on mycorrhizal fungi for seed germination in the wild. As part of a conservation plan for the species, research on seed germination is necessary. However, the molecular mechanism of seed germination and underlying orchid-fungus interactions during symbiotic germination are poorly understood. In this study, Illumina HiSeq 4000 transcriptome sequencing was performed to generate a substantial sequence dataset of germinating A. roxburghii seed. A mean of 44,214,845 clean reads were obtained from each sample. 173,781 unigenes with a mean length of 653 nt were obtained. A total of 51,514 (29.64%) sequences were annotated, among these, 49 unigenes encoding proteins involved in GA-GID1-DELLA regulatory module, including 31 unigenes involved in GA metabolism pathway, 5 unigenes encoding GID1, 11 unigenes for DELLA and 2 unigenes for GID2. A total of 11,881 genes showed significant differential expression in the symbiotic germinating seed sample compared with the asymbiotic germinating seed sample, of which six were involved in the GA-GID1-DELLA regulatory module, and suggested that they might be induced or suppressed by fungi. These results will help us understand better the molecular mechanism of orchid seed germination and orchid-fungus symbiosis.

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