For integers k and n with 2 ≤ k ≤ n − 1, a graph G of order n is k-path pancyclic if every path P of order k in G lies on a cycle of every length from k + 1 to n. Thus a 2-path pancyclic graph is edge-pancyclic. In this paper, we present sufficient conditions for graphs to be k-path pancyclic. For a graph G of order n ≥ 3, we establish sharp lower bounds in terms of n and k for (a) the minimum degree of G, (b) the minimum degree-sum of nonadjacent vertices of G and (c) the size of G such that G...

In a manner analogous to a commutative ring, the L-ideal-based L-zero-divisor graph of a commutative ring R can be defined as the undirected graph Γ(μ) for some L-ideal μ of R. The basic properties and possible structures of the graph Γ(μ) are studied.

A planar 3-connected graph G is essentially 4-connected if, for any 3-separator S of G, one component of the graph obtained from G by removing S is a single vertex. Jackson and Wormald proved that an essentially 4-connected planar graph on n vertices contains a cycle C such that [...] . For a cubic essentially 4-connected planar graph G, Grünbaum with Malkevitch, and Zhang showed that G has a cycle on at least ¾ n vertices. In the present paper the result of Jackson and Wormald is improved. Moreover,...

By a signpost system we mean an ordered pair $(W,P)$, where $W$ is a finite nonempty set, $P\subseteq W\times W\times W$ and the following statements hold: $$\text{if}(u,v,w)\in P,\text{then}(v,u,u)\in P\text{and}(v,u,w)\notin P,\text{for}\text{all}u,v,w\in W;\text{if}u\ne v,i\text{then}\text{there}\text{exists}r\in W\text{such}\text{that}(u,r,v)\in P,\text{for}\text{all}u,v\in W.$$ We say that a signpost system $(W,P)$ is smooth if the folowing statement holds for all $u,v,x,y,z\in W$: if $(u,v,x),(u,v,z),(x,y,z)\in P$, then $(u,v,y)\in P$. We say thay a signpost system $(W,P)$ is simple if the following statement holds for all $u,v,x,y\in W$: if $(u,v,x),(x,y,v)\in P$, then $(u,v,y),(x,y,u)\in P$. By the underlying graph of a signpost system $(W,P)$ we mean the graph $G$ with $V\left(G\right)=W$ and such that the following statement holds for all distinct $u,v\in W$: $u$ and $v$ are adjacent in $G$ if and only if $(u,v,v)\in P$....

In this paper we show bounds for the adjacent eccentric distance sum of graphs in terms of Wiener index, maximum degree and minimum degree. We extend some earlier results of Hua and Yu [Bounds for the Adjacent Eccentric Distance Sum, International Mathematical Forum. Vol. 7 (2O02) no. 26. 1280-1294]. The adjaceni eccentric distance sum index of the graph G is defined as [...] where ε(υ) is the eccentricity of the vertex υ, deg(υ) is the degree of the vertex υ and D(υ) = ∑u∊v(G) d (u,υ)is the sum...

We use an algebraic method to classify the generalized permutation star-graphs, and we use the classification to determine the toughness of all generalized permutation star-graphs.

In this note we show that $1$-skeletons and $2$-skeletons of $n$-pseudomanifolds with full boundary are $(n+1)$-connected graphs and $n$-connected $2$-complexes, respectively. This generalizes previous results due to Barnette and Woon.

An axiomatic characterization of the distance function of a connected graph is given in this note. The triangle inequality is not contained in this characterization.