14 research outputs found
Laplacian spectral properties of signed circular caterpillars
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
Signed bicyclic graphs with minimal index
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
On the average eccentricity‎, ‎the harmonic index and the largest signless Laplacian eigenvalue of a graph
The eccentricity of a vertex is the maximum distance from it to‎ ‎another vertex and the average eccentricity of a‎ ‎graph is the mean value of eccentricities of all vertices of‎ ‎‎. ‎The harmonic index of a graph is defined‎ ‎as the sum of over all edges of‎ ‎‎, ‎where denotes the degree of a vertex in ‎. ‎In‎ ‎this paper‎, ‎we determine the unique tree with minimum average‎ ‎eccentricity among the set of trees with given number of pendent‎ ‎vertices and determine the unique tree with maximum average‎ ‎eccentricity among the set of -vertex trees with two adjacent‎ ‎vertices of maximum degree ‎, ‎where ‎. ‎Also‎, ‎we‎ ‎give some relations between the average eccentricity‎, ‎the harmonic‎ ‎index and the largest signless Laplacian eigenvalue‎, ‎and strengthen‎ ‎a result on the Randi'{c} index and the largest signless Laplacian‎ ‎eigenvalue conjectured by Hansen and Lucas cite{hl}‎
Unsolved Problems in Spectral Graph Theory
Spectral graph theory is a captivating area of graph theory that employs the
eigenvalues and eigenvectors of matrices associated with graphs to study them.
In this paper, we present a collection of topics in spectral graph theory,
covering a range of open problems and conjectures. Our focus is primarily on
the adjacency matrix of graphs, and for each topic, we provide a brief
historical overview.Comment: v3, 30 pages, 1 figure, include comments from Clive Elphick, Xiaofeng
Gu, William Linz, and Dragan Stevanovi\'c, respectively. Thanks! This paper
will be published in Operations Research Transaction