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The quadrilateral graph Q(G) is obtained from G by replacing each edge in G with two parallel paths of length 1 and 3, whereas the pentagonal graph W(G) is obtained from G by replacing each edge in G with two parallel paths of length 1 and 4. In this paper, closed-form formulas of resistance distance and Kirchhoff index for quadrilateral graph and pentagonal graph are obtained whenever G is an arbitrary graph.
The resistance between two nodes in some resistor networks has been studied extensively by mathematicians and physicists. Let $L_n$ be a linear hexagonal chain with $n$, 6-cycles. Then identifying the opposite lateral edges of $L_n$ in ordered way yi
Xiong and Liu [L. Xiong and Z. Liu, Hamiltonian iterated line graphs, Discrete Math. 256 (2002) 407-422] gave a characterization of the graphs $G$ for which the $n$-th iterated line graph $L^n(G)$ is hamiltonian, for $nge2$. In this paper, we study t
A graph $G$ is $k$-path-coverable if its vertex set $V(G)$ can be covered by $k$ or fewer vertex disjoint paths. In this paper, using the $Q$-index of a connected graph $G$, we present a tight sufficient condition for $G$ with fixed minimum degree and large order to be $k$-path-coverable.
Let $ Pi_q $ be the projective plane of order $ q $, let $psi(m):=psi(L(K_m))$ the pseudoachromatic number of the complete line graph of order $ m $, let $ ain { 3,4,dots,tfrac{q}{2}+1 } $ and $ m_a=(q+1)^2-a $. In this paper, we improve the upper
We review the theory of Cheeger constants for graphs and quantum graphs and their present and envisaged applications.