The inertia and energy gap of a vertex-decorated graph with identically weighted ‘internal’ edges and beyond

Abstract

Let $G=(V, E)$ be an undirected graph with the vertex set V and the edge set E $(|V|=n, |E|=m)$. The spectrum of eigenvalues of G is the (multi)set of all eigenvalues $\lambda _{j}$ $(\lambda _{1}\ge \lambda _{2}\ge \cdots \ge \lambda _{n})$ of its adjacency matrix $A=[a_{jk}]_{j,k=1}^{n}$. The inertia (set or indices) of the matrix A (graph G) is the triple $\{n_{+}, n_{-}, n_{0}\}$, where $n_{+}, n_{-}, n_{0}$ are the numbers of positive, negative, and zero eigenvalues of A (G), respectively. Let $d_{1}, d_{2}, \ldots , d_{n}$ be a vertex degree sequence of G, and let ${\mathcal {H}}=\{H_{1}, H_{2}, \ldots , H_{n}\}$ be a (multi)set of connected undirected graphs with $|V(H_{j})|=d_{j}$ $(j\in \{1, 2, \ldots , n\})$. We consider the complete vertex decoration $G({\mathcal {H}})$ of G using the graphs $H_{j}$, when each vertex $v_{j}$ of G is decorated (replaced) by the respective graph $H_{j}$ with the vertex set $\{u_{1}, u_{2}, \ldots , u_{d_{j}}\}$. In the vertex-decorated graph $G({\mathcal {H}})$, each vertex $u_{j}$ of $H_{j}$ serves as a new $v_{j}$-end vertex for the former edge $v_{j}v_{k}$ of G. Considering an edge as a pair of opposite arcs with common endpoints, we determine how the inertia and energy gap of $G({\mathcal {H}})$ depend on the attachment of the same weight w to all arcs of the ${\mathcal {H}}$-edges of $G({\mathcal {H}})$. Some further generalizations and possible links to problems in chemistry are also briefly discussed.