
A graphic sequence
We show, using a hybrid analysis/linear algebra argument, that the diagonal vector of an infinite symmetric matrix over
Given a distribution of pebbles on the vertices of a connected graph
The enhanced hypercube is basically a hypercube with additional edges augmented, where the additional edges connect all pairs of complementary nodes in the hypercube. Taking into account the minimal routing function and the structural properties of the enhanced hypercube,
The newly introduced neighborhood matrix extends the power of adjacency and distance matrices to describe the topology of graphs. The adjacency matrix enumerates which pairs of vertices share an edge and it may be summarized by the degree sequence, a list of the adjacency matrix row sums. The distance matrix shows more information, namely the length of shortest paths between vertex pairs. We introduce and explore the neighborhood matrix, which we have found to be an analog to the distance matrix what the degree sequence is to the adjacency matrix. The neighbor matrix includes the degree sequence as its first column and the sequence of all other distances in the graph up to the graph’s diameter, enumerating the number of neighbors each vertex has at every distance present in the graph. We prove this matrix to contain eleven oft-used graph statistics and topological descriptors. We also provide insight into two applications that show potential utility of the neighbor matrix in comparing graphs and identifying topologically significant vertices in a graph.
In this paper, for the Catalan-Larcombe-French sequence
A graph
Given a large finite point set,
An explicit formula for the number of spanning trees of the lexi- cographic product GLH] of two arbitrary graphs G and H is deduced in terms of structure parameters of G and H. Some properties on the number of spanning trees of G[H] are revealed. Sharp lower and upper bounds for the number of spanning trees of lexicographic product of graphs are established. In particular, simple formulae for the number of spanning trees of the lexicographic product of some special graphs are derived, which extend some previously known results
in the literature.
For a graph
are considered, where
With the help of a computer, we show that
if
We also obtain the bounds
if
The diameter of a graph can be affected by the addition or the deletion of some edges. In [3], we have studied the diameter variability of the Cartesian product of graphs. In this paper, we discuss about two fundamental products, strong and lexicographic products of graphs, whose diameter increases (decreases) by the deletion (addition) of a single edge. The problems of minimality and maximality of the product graphs with respect to its diameter are also solved. These problems are motivated by the fact that these graph products are good interconnection networks.
The concept of the skew energy of a digraph was introduced by Adiga, Balakrishnan and So in 2010. An oriented graph
In this paper, according to the symmetric Lanczos algorithm and general Gauss-type quadrature rule, we give some lower bounds on the Resolvent Estrada index
The chordal-(
An odd open dominating set of a graph is a subset of the graph’s vertices with the property that the open neighborhood of each vertex in the graph contains an odd number of vertices in the subset. An odd closed
We show that the
Multireceiver authentication codes allow one sender to construct an authenticated message for a group of receivers such that each receiver can verify the authenticity of the received message. In this paper, we construct one multireceiver authentication code from pseudo-symplectic geometry over finite fields. The parameters and the probabilities of deceptions of the codes are also computed. The smaller the probability of successful attack, the higher the security of the authentication codes.
Ryser’s Conjecture states that for any
It is interesting to study hypergraphs which are extremal in Ryser’s Conjecture, i.e., those hypergraphs for which the vertex cover number is exactly
In this paper, we focus on the cases when
Let
A graph is said to be claw-free if it does not contain an induced subgraph isomorphic to
A graph
A star forest is a forest each of whose components is a star. The star arboricity of a graph
An adjacent vertex distinguishing total coloring of a graph
The concept of the skew energy of a digraph was introduced by Adiga, Balakrishnan and
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