Let qmin(G) stand for the smallest eigenvalue of the signless Laplacian of a graph G of order n: This paper gives some results on the following extremal problem: How large can qmin (G) be if G is a graph of order n; with no complete subgraph of order r + 1? It is shown that this problem is related to the well-known topic of making graphs bipartite. Using known classical results, several bounds ...

Define the neighborhood characteristic of a graph to be s1 − s2 + s3 − · · ·, where si counts subsets of i vertices that are all adjacent to some vertex outside the subset. This amounts to replacing cliques by neighborhoods in the traditional ‘Euler characteristic’ (the number of vertices, minus the number of edges, plus the number of triangles, etc.). The neighborhood characteristic can also b...

We study combinatorial aspects of the sandpile model on wheel and fan graphs, seeking bijective characterisations model’s recurrent configurations these families. For we exhibit a bijection between set subgraphs cycle graph which maps level configuration to number edges subgraph. This relies two key ingredients. The first consists in considering stochastic variant standard Abelian (ASM), rather...