Maxima of the Q-index: forbidden 4-cycle and 5-cycle

This paper gives tight upper bounds on the largest eigenvalue q(G) of the signless Laplacian of graphs with no 4-cycle and no 5-cycle. If n is odd, let F_{n} be the friendship graph of order n; if n is even, let F_{n} be F_{n-1} with an edge hanged to its center. It is shown that if G is a graph of order n, with no 4-cycle, then q(G)<q(F_{n}), unless G=F_{n}. Let S_{n,k} be the join of a complete graph of order k and an independent set of order n-k. It is shown that if G is a graph of order n, with no 5-cycle, then q(G)<q(S_{n,2}), unless G=S_{n,k}. It is shown that these results are significant in spectral extremal graph problems. Two conjectures are formulated for the maximum q(G) of graphs with forbidden cycles.Comment: 12 page

New lower bounds for the Randić spread

Let $G=\left( \mathcal{V}\left( G\right) ,\mathcal{E}\left( G\right) \right)$ be an $\left( n,m\right)$-graph. The Randi\'{c} spread of $G$, $s_{R}(G)$, is defined as the maximum distance of its Randi\'{c} eigenvalues, disregarding the Randi\'{c} spectral radius of $G$. In this work, we use numerical inequalities and bounds for the matricial spread to obtain relations between this spectral parameter and some structural and algebraic parameters of the underlying graph such as, the sequence of vertex degrees, the nullity, Randi\'{c} index, generalized Randi\'{c} indices and its independence number. In the last section a comparison is presented for regular graphs

Spectra, signless Laplacian and Laplacian spectra of complementary prisms of graphs

The complementary prism GG‾ of a graph G is obtained from the disjoint union of G and its complement G‾ by adding an edge for each pair of vertices (v,v′), where v is in G and its copy v′ is in G‾. The Petersen graph C5C5‾ and, for n≥2, the corona product of Kn and K1 which is KnKn‾ are examples of complementary prisms. This paper is devoted to the computation of eigenpairs of the adjacency, signless Laplacian and Laplacian matrices of a complementary prism GG‾ in terms of the eigenpairs of the corresponding matrices of G. Particular attention is given to the complementary prisms of regular graphs. Furthermore, Petersen graph is shown to be the unique complementary prism which is a strongly regular graph

Maxima of the Q-index: Graphs with no Ks,t

This note presents a new spectral version of the graph Zarankiewicz problem: How large can be the maximum eigenvalue of the signless Laplacian of a graph of order n that does not contain a specified complete bipartite subgraph. A conjecture is stated about general complete bipartite graphs, which is proved for infinitely many cases. More precisely, it is shown that if G is a graph of order n, with no subgraph isomorphic to K2,s+1, then the largest eigenvalue q(G) of the signless Laplacian of G satisfiesq(G) (Formula presented.), with equality holding if and only if G is a join of K1 and an s-regular graph of order n-1

Maxima of the Q-index: Forbidden 4-cycle and 5-cycle

This paper gives tight upper bounds on the largest eigenvalue q (G) of the signless Laplacian of graphs with no 4-cycle and no 5-cycle. If n is odd, let Fn be the friendship graph of order n; if n is even, let Fn be Fn-1 with an extra edge hung to its center. It is shown that if G is a graph of order n ≥ 4, with no 4-cycle, then q (G) \u3c q (Fn), unless G = Fn. Let Sn,k be the join of a complete graph of order k and an independent set of order n - k. It is shown that if G is a graph of order n ≥ 6, with no 5-cycle, then q (G) \u3c q (Sn,2), unless G = Sn,k. It is shown that these results are significant in spectral extremal graph problems. Two conjectures are formulated for the maximum q (G) of graphs with forbidden cycles

An overview on Randic (Normalized Laplacian) spread

The definition of Randic matrix comes from a molecular structure descriptor introduced by Milan Randic in 1975, known as Randic index. The plethora of chemical and pharmacological applications of the Randic index, as well as numerous mathematical investigations are well known and presented in the literature. In spite of its connection with Randic index this matrix seems to have not been much studied in mathematical chemistry however, some graph invariants related with this matrix such as Randic energy (the sum of the absolute values of the eigenvalues of the Randic matrix), the concept of Randic spread (that is, the maximum difference between two eigenvalues of the Randic matrix, disregarding the spectral radius) were recently introduced and some of their properties were established. We review some topics related with the graph invariant Randic spread, such as bounds that were obtained from matrix and/or numerical inequalities establishing relations between this spectral parameter and some structural parameters of the underlying graph. Moreover, some new bounds for the Randic spread are obtained. Comparisons with some upper bounds for the Randic spread of regular graphs are done. Finally, a possible relation between Randic spread and Randic energy is established