Non-bipartiteness of graphs and the upper bounds of Dirichlet forms (Q854743)

Let \(G\) be a graph and let \(V (G)\) (resp., \(E (G)\)) denote the set of its vertices (resp., its unoriented edges). It is assumed that the graph \(G\) is connected and locally finite and that it has more than one edge. Assigning two orientations to each edge in \(E(G)\), the authors denote by \(A(G)\) the set of all oriented edges. For an oriented edge \(e\in A (G)\), the origin vertex of \(e\), the terminal vertex of \(e\) and the inverse edge of \(e\) are denoted by \(o(e)\), \(t(e)\) and \(\overline{e}\), respectively. Let \(p:A(G)\to (0,1]\) be a positive transition probability on \(G\) so that \(\sum_{e\in A_{x}(G)}(e)=1\), where \(A_{x}(G)=\{e\in A(G):o(e)=x\}\). It is assumed that the positive transition probability \(p\) is inversible, meaning that there exists a positive valued functions \(m:V(G)\to(0,\infty)\) (reversible measure) such that \(m(o(e))p(e)=m (t(e))p(\overline{e})\) for every oriented edge \(e\in A(G)\). Further, let \(\ell^{2}(G,m)\) denote the Hilbert space of all functions \(f:V(G)\to\mathbb{R}\) such that \(\| f \|_{m}^{2}:=\langle f,f \rangle_{m}<\infty\), where \(\langle\cdot,\cdot\rangle_{m}\) denotes the scalar product on \(\ell^{2} (G,m)\) defined by \(\langle f_{1},f_{2} \rangle_{m} = \sum_{x \in V (G)} f_{1}(x) f_{2}(x) m(x)\), \(f_{1},f_{2} \in l^{2} (G,m)\). On the space \(\ell^{2}(G,m)\), define the discrete Laplacian \(\Delta_{n}\) associated with the transition probability \(p\) by \[ \Delta_{p}f(x)=\sum_{e\in A_{x}(G)}p(e)df(e), \] where \(df(e)=f(t(e))-f(o(e))\). The corresponding Dirichlet form \(\mathcal{E}_{p}\) on \(\ell^{2}(G,m)\) is defined by \[ \mathcal{E}_{p} (f,f) = \langle -\Delta_{p} f,f \rangle_{m},\;f \in l^{2} (G,m). \] \(-\Delta_{p}\) is a bounded self-adjoint operator on \(\ell^{2}(G,m)\) and its spectrum is a closed subset in \([0,2]\). Let \(\lambda_{0}(p)\) and \(\lambda_{\infty}(p)\) be the lower bound and the upper bound of the spectrum of \(-\Delta_{p}\), respectively. So, \(\lambda_{0}(p):=\inf\{\mathcal{E}_{p}(f,f):\| f \|_{m}=1\}\) and \(\lambda_{\infty}(p):= \sup\{\mathcal{E}_{p}(f,f):\| f \|_{m}=1\}\). The main result of the present paper is the following Theorem. If \(G\) is an essentially non-bipartite graph, then \(\lambda_{0}(p)+\lambda_{\infty}(p)< 2\), whenever \(\inf_{e\in A(G)}p(e)>0\).