The spectrum of the Hilbert space valued second derivative with general self-adjoint boundary conditions

We consider a large class of self-adjoint elliptic problem associated with the second derivative acting on a space of vector-valued functions. We present two different approaches to the study of the associated eigenvalues problems. The first, more general one allows to replace a secular equation (which is well-known in some special cases) by an abstract rank condition. The latter seems to apply particularly well to a specific boundary condition, sometimes dubbed "anti-Kirchhoff" in the literature, that arise in the theory of differential operators on graphs; it also permits to discuss interesting and more direct connections between the spectrum of the differential operator and some graph theoretical quantities. In either case our results yield, among other, some results on the symmetry of the spectrum

Algebraic matroids with graph symmetry

This paper studies the properties of two kinds of matroids: (a) algebraic matroids and (b) finite and infinite matroids whose ground set have some canonical symmetry, for example row and column symmetry and transposition symmetry. For (a) algebraic matroids, we expose cryptomorphisms making them accessible to techniques from commutative algebra. This allows us to introduce for each circuit in an algebraic matroid an invariant called circuit polynomial, generalizing the minimal poly- nomial in classical Galois theory, and studying the matroid structure with multivariate methods. For (b) matroids with symmetries we introduce combinatorial invariants capturing structural properties of the rank function and its limit behavior, and obtain proofs which are purely combinatorial and do not assume algebraicity of the matroid; these imply and generalize known results in some specific cases where the matroid is also algebraic. These results are motivated by, and readily applicable to framework rigidity, low-rank matrix completion and determinantal varieties, which lie in the intersection of (a) and (b) where additional results can be derived. We study the corresponding matroids and their associated invariants, and for selected cases, we characterize the matroidal structure and the circuit polynomials completely

Characterizing Block Graphs in Terms of their Vertex-Induced Partitions

Given a finite connected simple graph $G=(V,E)$ with vertex set $V$ and edge set $E\subseteq \binom{V}{2}$, we will show that $1.$ the (necessarily unique) smallest block graph with vertex set $V$ whose edge set contains $E$ is uniquely determined by the $V$-indexed family ${\bf P}_G:=\big(\pi_0(G^{(v)})\big)_{v \in V}$ of the various partitions $\pi_0(G^{(v)})$ of the set $V$ into the set of connected components of the graph $G^{(v)}:=(V,\{e\in E: v\notin e\})$, $2.$ the edge set of this block graph coincides with set of all $2$-subsets $\{u,v\}$ of $V$ for which $u$ and $v$ are, for all $w\in V-\{u,v\}$, contained in the same connected component of $G^{(w)}$, $3.$ and an arbitrary $V$-indexed family ${\bf P}p=({\bf p}_v)_{v \in V}$ of partitions $\pi_v$ of the set $V$ is of the form ${\bf P}p={\bf P}p_G$ for some connected simple graph $G=(V,E)$ with vertex set $V$ as above if and only if, for any two distinct elements $u,v\in V$, the union of the set in ${\bf p}_v$ that contains $u$ and the set in ${\bf p}_u$ that contains $v$ coincides with the set $V$, and $\{v\}\in {\bf p}_v$ holds for all $v \in V$. As well as being of inherent interest to the theory of block graphs, these facts are also useful in the analysis of compatible decompositions and block realizations of finite metric spaces

Generalized chordality, vertex separators and hyperbolicity on graphs

Let $G$ be a graph with the usual shortest-path metric. A graph is $\delta$-hyperbolic if for every geodesic triangle $T$, any side of $T$ is contained in a $\delta$-neighborhood of the union of the other two sides. A graph is chordal if every induced cycle has at most three edges. A vertex separator set in a graph is a set of vertices that disconnects two vertices. In this paper we study the relation between vertex separator sets, some chordality properties which are natural generalizations of being chordal and the hyperbolicity of the graph. We also give a characterization of being quasi-isometric to a tree in terms of chordality and prove that this condition also characterizes being hyperbolic, when restricted to triangles, and having stable geodesics, when restricted to bigons.Comment: 16 pages, 3 figure