Let r(G) be the minimum number of complete bipartite subgraphs needed to partition the edges of G, and let r'G) be the larger of the number of positive and number of negative eigenvalues of G. It is known that T{G) > r(G); graphs with t(G) = r(G) are called eigensharp. Eigensharp graphs include graphs, trees, cycles Cn with n = 4 or n ^ 4k, prisms Cn\2K2 with n ^ 3fc, "twisted prisms" (also cal...

An edge e of a simple 3-connected graph G is essential if neither the deletion G\e nor the contraction G/e is both simple and 3–connected. Tutte’s Wheels Theorem proves that the only simple 3–connected graphs with no non-essential edges are the wheels. In earlier work, as a corollary of a matroid result, the authors determined all simple 3-connected graphs with at most two non-essential edges. ...

We consider a PT-symmetric partner to Khare-Mandal’s recently proposed non-Hermitian potential with complex eigenvalues. Our potential, which is quasi-exactly solvable, is shown to possess only real eigenvalues. PACS : 03.65.Bz, 03.65.Ge

Let $$G_S$$ be a graph with loops obtained from G of order n and at $$S \subseteq V(G).$$ In this paper, we establish neccesary sufficient condition on the bipartititeness connected spectrum $${\textrm{Spec}}(G_S)$$ $${\textrm{Spec}}(G_{V(G)\backslash S})$$ . We also prove that for every $$S\subseteq V(G),$$ $${\mathcal E}(G_S) \ge {\mathcal E}(G)$$ when is bipartite. Moreover, provide an ident...

The reciprocal complementary distance (RCD) matrix of a graphG is defined asRCD(G) = [rcij] where rcij = 1 1+D−dij if i 6= j and rcij = 0, otherwise, where D is the diameter of G and dij is the distance between the vertices vi and vj in G. The RCD-energy of G is defined as the sum of the absolute values of the eigenvalues of RCD(G). Two graphs are said to be RCDequienergetic if they have same R...

the sharp upper bounds and the sharp lower bounds of the largest eigenvalues $lambda_1$, the least eigenvalue $lambda_n$, the second largest eigenvalue $lambda_2$, the spread and the separator among all firefly graphs on $n$ vertices are determined.