Let $K_{m}-H$ be the graph obtained from $K_{m}$ by removing the edges set $E(H)$ of the graph $H$ ($H$ is a subgraph of $K_{m}$). We use the symbol $Z_4$ to denote $K_4-P_2.$ A sequence $S$ is potentially $K_{m}-H$-graphical if it has a realization containing a $K_{m}-H$ as a subgraph. Let $\sigma(K_{m}-H, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $\sigma(S)\geq \sigma(K_{m}-H, n)$ is potentially $K_{m}-H$-graphical. In this paper, we determine the values of $\sigma (K_{r+1}-Z, n)$ for $n\geq 5r+19, r+1 \geq k \geq 5,$ $j \geq 5$ where $Z$ is a graph on $k$ vertices and $j$ edges which contains a graph $Z_4$ but not contains a cycle on $4$ vertices. We also determine the values of $\sigma (K_{r+1}-Z_4, n)$, $\sigma (K_{r+1}-(K_4-e), n)$, $\sigma (K_{r+1}-K_4, n)$ for $n\geq 5r+16, r\geq 4$.

For any r-order {0,1}-tensor A with e ones, we prove that the spectral radius of A is at most exp((r-1)/r) with the equality holds if and only if e=k^r for some integer k and all ones forms a principal sub-tensor 1_{k x k x ... x k}. We also prove a stability result for general tensor A with e ones where e = k^r+l with relatively small l. Using the stability result, we completely characterized the tensors achieving the maximum spectral radius among all r-order {0,1}-tensor A with k^r+l ones, for -r-1 ≤ l ≤ r, and k sufficiently large.

We prove a homology vanishing theorem for graphs with positive Bakry-\'Emery curvature, analogous to a classic result of Bochner on manifolds\cite {Bochner}. Specifically, we prove that if a graph has positive curvature at every vertex, then its first homology group is trivial, where the notion of homology that we use for graphs is the path homology developed by Grigor'yan, Lin, Muranov, and Yau\cite {Grigoryan2}.%\Hm {added the fundamental group curvature relation} We moreover prove that the fundamental group is finite for graphs with positive Bakry-\'Emery curvature, analogous to a classic result of Myers on manifolds\cite {Myers1941}. The proofs draw on several separate areas of graph theory. We study graph coverings, gain graphs, and cycle spaces of graphs, in addition to the Bakry-\'Emery curvature and the path homology. The main results follow as a consequence of several different relationships developed among these different areas. Specifically, we show that a graph with positive curvature can have no non-trivial infinite cover preserving 3-cycles and 4-cycles, and give a combinatorial interpretation of the first path homology in terms of the cycle space of a graph. We relate cycle spaces of graphs to gain graphs with abelian gain group, and relate these to coverings of graphs. Along the way, we prove other new facts about gain graphs, coverings, and cycles spaces that are of related interest. Furthermore, we relate gain graphs to graph homotopy and the fundamental group developed by Grigor'yan, Lin, Muranov, and Yau\cite {Grigoryan_homotopy}, and obtain an alternative proof to their result that the abelianization of the fundamental

We construct small cancellation labellings for some infinite sequences of finite graphs of bounded degree. We use them to define infinite graphical small cancellation presentations of groups. This technique allows us to provide examples of groups with exotic properties: • We construct the first examples of finitely generated coarsely non-amenable groups (that is, groups without Guoliang Yu’s Property A) that are coarsely embeddable into a Hilbert space. Moreover, our groups act properly on CAT(0) cubical complexes. • We construct the first examples of finitely generated groups, with expanders embedded isometrically into their Cayley graphs—in contrast, in the case of the Gromov monster expanders are not even coarsely embedded. We present further applications.

Geometric genus is an important invariant in the classification theory for isolated singularities. In this paper we give a complete classification of three-dimensional isolated weighted homogeneous singularities with geometric genus one. This is one of important classes of minimally elliptic singularities.