Let G=(V(G),E(G)) be a simple connected unweighted graph. A set R⊂V(G) is called fault-tolerant resolving with the tolerance level k if cardinality of Sx,y={w∈R:d(w,x)≠d(w,y)} at least for every pair distinct vertices x,y G. k-level metric dimension refers to minimum size k. In this article, we calculate and determine circulant graph C(n:1,2) all possible values n. The optimal sets are also del...

Extremal problems in graph theory form a very wide research area. We study the following topics: the metric dimension of circulant graphs, the Wiener index of trees of given diameter, and the degree‐diameter problem for Cayley graphs. All three topics are connected to the study of distances in graphs. We give a short survey on the topics and present several new results.

We demonstrate that a quantum graph exhibits $\mathcal{PT}$-symmetry provided the coefficients in condition describing wave function matching at vertices are circulant matrices; this symmetry is nontrivial if they not invariant with respect to transposition. also illustrate how transport properties of such graphs significantly influenced by presence or absence non-Robin component coupling.

Recent advances in graph theory, linear algebra, and commutative algebra render us to tackle problems one bough of mathematics with assistance guidance from others. We will elaborate foremost conceptually fathomless homological invariants inextricably linked circulant matrices cycles through various path lengths this article, as well a class Koszul which portrays combinatorial correlation, the ...

We construct a deterministic algorithm that tests whether two circulant graphs are isomorphic. The running time is O(n2(logn)6), where n is the number of vertices of each graph. Our algorithm works for directed, undirected, and edge-colored circulants.

‎the subject of this paper is the solution of the fredholm integral equation with toeplitz, hankel and the toeplitz plus hankel kernel. the mean value theorem for integrals is applied and then extended for solving high dimensional problems and finally, some example and graph of error function are presented to show the ability and simplicity of the ‎method.

The cyclicity of a hypergraph is an e ciently computable integer that extends the notion of the cyclomatic number of a graph The formula for the cyclicity is suggested by the join invariant of an acyclic hypergraph which is the multiset of all joining sets in any of its join trees Once we gure out how the multiplicity of a joining set depends on the structure of the acyclic hypergraph we de ne ...