Non-Singular graphs with a Singular Deck

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• 作者：Alexander Farrugia ^{alex.farrugia@um.edu.mt" class="auth_mail" title="E-mail the corresponding author} ; John Baptist Gauci ^{john-baptist.gauci@um.edu.mt" class="auth_mail" title="E-mail the corresponding author} ; Irene Sciriha ; ^{isci1@um.edu.mt" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Adjacency matrix ; Singular graph ; NSSD ; G-nutful graph ; Vertex-deleted subgraphs ; SSP model
• 刊名：Discrete Applied Mathematics
• 出版年：2016
• 出版时间：31 March 2016
• 年：2016
• 卷：202
• 期：Complete
• 页码：50-57
• 全文大小：465 K

文摘

The n$n$-vertex graph $G(=\Gamma \left(G\right))$ with a non-singular real symmetric adjacency matrix $G$, having a zero diagonal and singular (n−1)×(n−1)$(n-1)\times (n-1)$ principal submatrices is termed a NSSD, a Non-Singular graph with a Singular Deck. NSSDs arose in the study of the polynomial reconstruction problem and were later found to characterise non-singular molecular graphs that are distinct omni-conductors and ipso omni-insulators. Since both matrices $G$ and a0594419d4c3723c5b85f7fcb819">$G^{-1}$ represent NSSDs 08a22ba605ce73957d2dd08f">$\Gamma \left(G\right)$ and $\Gamma \left(G^{-1}\right)$, the value of the nullity of a one-, two- and three-vertex deleted subgraph of G$G$ is shown to be determined by the corresponding subgraph in $\Gamma \left(G^{-1}\right)$. Constructions of infinite subfamilies of non-NSSDs are presented. NSSDs with all two-vertex deleted subgraphs having a common value of the nullity are referred to as $G$-nutful graphs. We show that their minimum vertex degree is at least 4.