Let $f(n)$ be the maximum number of edges in a graph on $n$ vertices in which no two cycles have the same length. In 1975, Erdős raised the question of determining $f(n)$ [see J. A. Bondy and U. S. R. Murty, Graph theory with applications, Elsevier, New York, 1976; MR0411988 (54 #117)]. We prove that for $n\geq 36\cdot 5t^2-4\cdot 5t+1$ one has $f(n)\geq n+9t-1$ . We conjecture that $\liminf (f(n)-n)/\sqrt n\leq\sqrt 3$ .

We derive bounds for eigenvalues of the Laplacian of graphs using discrete versions of the Sobolev inequalities and heat kernel estimates.

We establish a new on-diagonal lower estimate of continuous-time heat kernels for large time on graphs. To achieve the goal, we first introduce an upper estimate of heat kernels in natural graph metric, then use the upper estimate and the volume growth condition to show the validity of the on-diagonal lower estimate.

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

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