## Graphs with the maximal Estrada indices.(English)Zbl 1292.05181

Summary: Two new transformations are proposed to compare the Estrada indices between two graphs. Let $${\Psi}_{n,m}$$ be the set of the $$(n,m)$$-graphs, where $$n$$ and $$m$$ are the numbers of vertices and edges, respectively. The graphs with the maximal Estrada indices in $${\Psi}_{n,m}$$ are deduced by the new method for three cases, namely unicyclic and bipartite unicyclic graphs $$(m=n)$$, bicyclic graphs $$(m=n+1)$$, and the $$(n,m)$$-graphs without even cycles $$(n+1 \leqslant m\leqslant 3(n-1)/2)$$.

### MSC:

 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 05C35 Extremal problems in graph theory

### Keywords:

$$(n, m)$$-graph; Estrada index; transformation
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