3 research outputs found

    On Energy, Laplacian Energy and PP-fold Graphs

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    For a graph GG having adjacency spectrum (AA-spectrum) Ξ»n≀λnβˆ’1≀⋯≀λ1\lambda_n\leq\lambda_{n-1}\leq\cdots\leq\lambda_1 and Laplacian spectrum (LL-spectrum) 0=ΞΌn≀μnβˆ’1≀⋯≀μ10=\mu_n\leq\mu_{n-1}\leq\cdots\leq\mu_1, the energy is defined as E(G)=βˆ‘i=1n∣λi∣ E(G)=\sum_{i=1}^{n}|\lambda_i| and the Laplacian energy is defined as LE(G)=βˆ‘i=1n∣μiβˆ’2mn∣LE(G)=\sum_{i=1}^{n}|\mu_i-\frac{2m}{n}|. In this paper, we give upper and lower bounds for the energy of KKnj,Β 1≀j≀nKK_n^j,~1\leq j \leq n and as a consequence we generalize a result of Stevanovic et al. [More on the relation between Energy and Laplacian energy of graphs, MATCH Commun. Math. Comput. Chem. {\bf 61} (2009) 395-401]. We also consider strong double graph and strong pp-fold graph to construct some new families of graphs GG for which E(G)> LE(G)

    On Scores, Losing Scores and Total Scores in Hypertournaments

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    A kk-hypertournament is a complete kk-hypergraph with each kk-edge endowed with an orientation, that is, a linear arrangement of the vertices contained in the edge. In a kk-hypertournament, the score sis_{i} (losing score rir_{i}) of a vertex viv_{i} is the number of arcs containing viv_{i} in which viv_{i} is not the last element (in which viv_{i} is the last element). The total score of viv_{i} is defined as ti=siβˆ’rit_{i}=s_{i}-r_{i}. In this paper we obtain stronger inequalities for the quantities βˆ‘i∈Iri\sum_{i\in I}r_{i}, βˆ‘i∈Isi\sum_{i\in I}s_{i} and βˆ‘i∈Iti\sum_{i\in I}t_{i}, where IβŠ†{1,2,…,n}I\subseteq \{ 1,2,\ldots,n\}. Furthermore, we discuss the case of equality for these inequalities. We also characterize total score sequences of strong kk-hypertournaments
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