Let [Formula: see text] be a graph. A set [Formula: see text] is a local resolving set of [Formula: see text] if there exists [Formula: see text] such that [Formula: see text] for any [Formula: see text]. The local metric dimension [Formula: see text] of [Formula: see text] is the minimum cardinality of all the local resolving sets of [Formula: see text]. In this paper, we investigate the local metric dimensions of several typical networks, including antiprism networks, hexagonal ladder networks, radix triangular mesh networks, and sierpiński networks.
The spectrum of a matrix refers to the multiset of its eigenvalues, and its spectral radius refers to the maximum absolute value of its eigenvalues. The energy of a matrix is defined as the sum of the absolute values of all its eigenvalues. For a connected graph [Formula: see text] with vertex set [Formula: see text], the (vertex-) edge-connectivity matrix of [Formula: see text] is a square matrix of order [Formula: see text] whose corresponding entry in row [Formula: see text] and column [Formula: see text] is the maximum number of (internally vertex-) edge-disjoint paths connecting [Formula: see text] and [Formula: see text] for [Formula: see text]; otherwise, the corresponding entry is 0. The spectra and energy of vertex-connectivity matrices of graphs have been considered extensively in the existing literature. Recently, Akbari et al. [On edge-path eigenvalues of graphs, Linear Multilinear Algebra, 70(15)(2022) 2998–3008.] further introduced the edge-connectivity matrix of a graph, and presented some properties of edge-connectivity matrices with respect to the edge-connectivity of several graph classes. In this paper, we determine the spectra of edge-connectivity matrices for several special graph classes, and obtain some properties of the edge-connectivity matrices of graphs and upper and lower bounds on the spectral radii of the edge-connectivity matrices for several graph classes. Moreover, we establish upper and lower bounds on the energy of edge-connectivity matrices of graphs, and further obtain the exact values of the energy of the edge-connectivity matrices for bicyclic graphs of Type [Formula: see text] and Cartesian products of several special graph classes.
Let [Formula: see text] be a connected graph, and let [Formula: see text] and [Formula: see text] be two positive integers with [Formula: see text] (mod 2). A [Formula: see text]-odd factor of [Formula: see text] is a spanning subgraph [Formula: see text] of [Formula: see text] with [Formula: see text] (mod 2) and [Formula: see text] for every [Formula: see text]. A graph [Formula: see text] is called [Formula: see text]-critical with respect to [Formula: see text]-odd factor if [Formula: see text] contains a [Formula: see text]-odd factor for every [Formula: see text] with [Formula: see text]. Let [Formula: see text] denote the distance matrix of [Formula: see text]. The largest eigenvalue of [Formula: see text], denoted by [Formula: see text], is called the distance spectral radius of [Formula: see text]. In this paper, we prove an upper bound for [Formula: see text] in a [Formula: see text]-connected graph [Formula: see text] which guarantees [Formula: see text] to be [Formula: see text]-critical with respect to [Formula: see text]-odd factor.
Let FF(e )denote the set of faulty edges in folded hypercube FQ(n) with n >= 3 and |FFe|<= 3n - 8. We prove that there exists a Hamiltonian cycle in FQ(n) - FFe if the following two conditions are satisfied: (1) every vertex in FQ(n) - FFe has degree at least 2, and (2) there is at most one vertex with degree 2 in FQ(n) - FFe.
In this paper, the authors study the concept of domination in the negation of Cayley signed graphs over finite abelian groups. They establish sharp bounds for the domination numbers and characterize the groups that attain these bounds. The study reveals that, for certain Cayley sets, the sum of the domination numbers of a Cayley signed graph and its negation exceeds the group's order. The results are illustrated with several examples.
