31 research outputs found

    The signless Laplacian spectral radius of bicyclic graphs with a given girth

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    On the maximum AαA_{\alpha}-spectral radius of unicyclic and bicyclic graphs with fixed girth or fixed number of pendant vertices

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    For a connected graph GG, let A(G)A(G) be the adjacency matrix of GG and D(G)D(G) be the diagonal matrix of the degrees of the vertices in GG. The AαA_{\alpha}-matrix of GG is defined as \begin{align*} A_\alpha (G) = \alpha D(G) + (1-\alpha) A(G) \quad \text{for any α∈[0,1]\alpha \in [0,1]}. \end{align*} The largest eigenvalue of Aα(G)A_{\alpha}(G) is called the AαA_{\alpha}-spectral radius of GG. In this article, we characterize the graphs with maximum AαA_{\alpha}-spectral radius among the class of unicyclic and bicyclic graphs of order nn with fixed girth gg. Also, we identify the unique graphs with maximum AαA_{\alpha}-spectral radius among the class of unicyclic and bicyclic graphs of order nn with kk pendant vertices.Comment: 16 page

    Laplacian spectral properties of signed circular caterpillars

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    A circular caterpillar of girth n is a graph such that the removal of all pendant vertices yields a cycle Cn of order n. A signed graph is a pair Γ = (G, σ), where G is a simple graph and σ ∶ E(G) → {+1, −1} is the sign function defined on the set E(G) of edges of G. The signed graph Γ is said to be balanced if the number of negatively signed edges in each cycle is even, and it is said to be unbalanced otherwise. We determine some bounds for the first n Laplacian eigenvalues of any signed circular caterpillar. As an application, we prove that each signed spiked triangle (G(3; p, q, r), σ), i. e. a signed circular caterpillar of girth 3 and degree sequence πp,q,r = (p + 2, q + 2, r + 2, 1,..., 1), is determined by its Laplacian spectrum up to switching isomorphism. Moreover, in the set of signed spiked triangles of order N, we identify the extremal graphs with respect to the Laplacian spectral radius and the first two Zagreb indices. It turns out that the unbalanced spiked triangle with degree sequence πN−3,0,0 and the balanced spike triangle (G(3; p, ^ q, ^ r^), +), where each pair in {p, ^ q, ^ r^} differs at most by 1, respectively maximizes and minimizes the Laplacian spectral radius and both the Zagreb indices

    On the positive and negative inertia of weighted graphs

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    The number of the positive, negative and zero eigenvalues in the spectrum of the (edge)-weighted graph GG are called positive inertia index, negative inertia index and nullity of the weighted graph GG, and denoted by i+(G)i_+(G), i−(G)i_-(G), i0(G)i_0(G), respectively. In this paper, the positive and negative inertia index of weighted trees, weighted unicyclic graphs and weighted bicyclic graphs are discussed, the methods of calculating them are obtained.Comment: 12. arXiv admin note: text overlap with arXiv:1107.0400 by other author

    Spectral properties of digraphs with a fixed dichromatic number

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    Eccentric connectivity index

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    The eccentric connectivity index ξc\xi^c is a novel distance--based molecular structure descriptor that was recently used for mathematical modeling of biological activities of diverse nature. It is defined as ξc(G)=∑v∈V(G)deg(v)⋅ϵ(v)\xi^c (G) = \sum_{v \in V (G)} deg (v) \cdot \epsilon (v)\,, where deg(v)deg (v) and ϵ(v)\epsilon (v) denote the vertex degree and eccentricity of vv\,, respectively. We survey some mathematical properties of this index and furthermore support the use of eccentric connectivity index as topological structure descriptor. We present the extremal trees and unicyclic graphs with maximum and minimum eccentric connectivity index subject to the certain graph constraints. Sharp lower and asymptotic upper bound for all graphs are given and various connections with other important graph invariants are established. In addition, we present explicit formulae for the values of eccentric connectivity index for several families of composite graphs and designed a linear algorithm for calculating the eccentric connectivity index of trees. Some open problems and related indices for further study are also listed.Comment: 25 pages, 5 figure

    Signed bicyclic graphs with minimal index

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    The index of a signed graph \Sigma = (G; \sigma) is just the largest eigenvalue of its adjacency matrix. For any n > 4 we identify the signed graphs achieving the minimum index in the class of signed bicyclic graphs with n vertices. Apart from the n = 4 case, such graphs are obtained by considering a starlike tree with four branches of suitable length (i.e. four distinct paths joined at their end vertex u) with two additional negative independent edges pairwise joining the four vertices adjacent to u. As a by-product, all signed bicyclic graphs containing a theta-graph and whose index is less than 2 are detected
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