211,992 research outputs found
Some inequalities involving the distance signless Laplacian eigenvalues of graphs
Given a simple graph G, the distance signless Laplacian DQ(G) = Tr(G) + D(G) is the sum of vertex transmissions matrix T r(G) and distance matrix D(G). In this paper, thanks to the symmetry of DQ(G), we obtain novel sharp bounds on the distance signless Laplacian eigenvalues of G, and in particular the distance signless Laplacian spectral radius. The bounds are expressed through graph diameter, vertex covering number, edge covering number, clique number, independence number, domination number as well as extremal transmission degrees. The graphs achieving the corresponding bounds are delineated. In addition, we investigate the distance signless Laplacian spectrum induced by Indu-Bala product, Cartesian product as well as extended double cover graph
The additive-multiplicative distance matrix of a graph, and a novel third invariant
Graham showed with Pollak and Hoffman-Hosoya that for any directed graph
with strong blocks , the determinant and cofactor-sum
of the distance matrix can be computed from the same
quantities for the blocks . This was extended to trees - and in our recent
work to any graph - with multiplicative and -distance matrices. For trees,
we went further and unified all previous variants with weights in a unital
commutative ring, into a distance matrix with additive and multiplicative
edge-data.
In this work: (1) We introduce the additive-multiplicative distance matrix
of every strongly connected graph , using what we term the
additive-multiplicative block-datum . This subsumes the previously
studied additive, multiplicative, and -distances for all graphs. (2) We
introduce an invariant that seems novel to date, and use it to
show "master" Graham-Hoffman-Hosoya (GHH) identities, which express in terms of the blocks . We show how these imply all previous
variants. (3) We show depend only on the
block-data for not just , but also several minors of . This was not
studied in any setting to date; we show it in the "most general"
additive-multiplicative setting, hence in all known settings. (4) We compute
in closed-form; this specializes to all known variants. In
particular, we recover our previous formula for for
additive-multiplicative trees (which itself specializes to a result of
Graham-Lovasz and answers a 2006 question of Bapat-Lal-Pati.) (5) We also show
that not the Laplacian, but a closely related matrix is the "correct" one to
use in - for the most general additive-multiplicative matrix
of each . As examples, we compute in closed form for hypertrees.Comment: 27 pages, LaTe
QESK: Quantum-based Entropic Subtree Kernels for Graph Classification
In this paper, we propose a novel graph kernel, namely the Quantum-based
Entropic Subtree Kernel (QESK), for Graph Classification. To this end, we
commence by computing the Average Mixing Matrix (AMM) of the Continuous-time
Quantum Walk (CTQW) evolved on each graph structure. Moreover, we show how this
AMM matrix can be employed to compute a series of entropic subtree
representations associated with the classical Weisfeiler-Lehman (WL) algorithm.
For a pair of graphs, the QESK kernel is defined by computing the
exponentiation of the negative Euclidean distance between their entropic
subtree representations, theoretically resulting in a positive definite graph
kernel. We show that the proposed QESK kernel not only encapsulates complicated
intrinsic quantum-based structural characteristics of graph structures through
the CTQW, but also theoretically addresses the shortcoming of ignoring the
effects of unshared substructures arising in state-of-the-art R-convolution
graph kernels. Moreover, unlike the classical R-convolution kernels, the
proposed QESK can discriminate the distinctions of isomorphic subtrees in terms
of the global graph structures, theoretically explaining the effectiveness.
Experiments indicate that the proposed QESK kernel can significantly outperform
state-of-the-art graph kernels and graph deep learning methods for graph
classification problems
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