Let $G$ be a graph with order $p$, size $q$ and component number $\omega $. For each $i$ between $p - \omega $ and $q$, let ${\mathcal C}_{i}(G)$ be the family of spanning $i$-edge subgraphs of $G$ with exactly $\omega $ components. For an integer-valued graphical invariant $\varphi $, if $H \rightarrow H^{\prime }$ is an adjacent edge transformation (AET) implies $|\varphi (H) - \varphi (H^{\prime })| \le 1$, then $\varphi $ is said to be continuous with respect to AET. Similarly define the continuity of $\varphi $ with respect to simple edge transformation (SET). Let $M_{j}(\varphi )$ and $m_{j}(\varphi )$ be the invariants defined by $M_{j}(\varphi )(H) = \max _{T \in {\mathcal C}_{j}(H)} \varphi (T)$, $ m_{j}(\varphi )(H) = \min _{T \in {\mathcal C}_{j}(H)} \varphi (T) $. It is proved that both $M_{p - \omega }(\varphi )$ and $m_{p - \omega }(\varphi )$ interpolate over $\mathbf{{\mathcal C}_{i}(G)}$, $ p - \omega \le i \le q$, if $\varphi $ is continuous with respect to AET, and that $M_{j}(\varphi )$ and $m_{j}(\varphi )$ interpolate over $\mathbf{{\mathcal C}_{i}(G)}$, $p - \omega \le j \le i \le q$, if $\varphi $ is continuous with respect to SET. In this way a lot of known interpolation results, including a theorem due to Schuster etc., are generalized.
The notion of functions dependent locally on finitely many coordinates plays an important role in the theory of smoothness and renormings on Banach spaces, especially when higher order smoothness is involved. In this note we investigate the structural properties of Banach spaces admitting (arbitrary) bump functions depending locally on finitely many coordinates.
In the present paper we give general nonuniqueness results which cover most of the known nonuniqueness criteria. In particular, we obtain a generalization of the nonuniqueness theorem of Chr. Nowak, of Samimi’s nonuniqueness theorem and of Stettner’s nonuniqueness criterion.
In a recent paper the authors proposed a lower bound on $1 - \lambda _i$, where $\lambda _i$, $ \lambda _i \ne 1$, is an eigenvalue of a transition matrix $T$ of an ergodic Markov chain. The bound, which involved the group inverse of $I - T$, was derived from a more general bound, due to Bauer, Deutsch, and Stoer, on the eigenvalues of a stochastic matrix other than its constant row sum. Here we adapt the bound to give a lower bound on the algebraic connectivity of an undirected graph, but principally consider the case of equality in the bound when the graph is a weighted tree. It is shown that the bound is sharp only for certain Type I trees. Our proof involves characterizing the case of equality in an upper estimate for certain inner products due to A. Paz.