On line graphs with maximum energy

Abstract

For an undirected simple graph G, the line graph L(G) is the graph whose vertex set is in one-to-one correspondence with the edge set of G where two vertices are adjacent if their corresponding edges in G have a common vertex. The energy E(G) is the sum of the absolute values of the eigenvalues of G. The vertex connectivity κ(G) is referred as the minimum number of vertices whose deletion disconnects G. The positive inertia ν+(G) corresponds to the number of positive eigenvalues of G. Finally, the matching number β(G) is the maximum number of independent edges of G. In this paper, we derive a sharp upper bound for the energy of the line graph of a graph G on n vertices having a vertex connectivity less than or equal to k. In addition, we obtain upper bounds on E(G) in terms of the edge connectivity, the inertia and the matching number of G. Moreover, a new family of hyperenergetic graphs is obtained.