Sentence examples for be the unique cycle from inspiring English sources

Let (C_{l}) be the unique cycle of G of length l.

Lemma 4.6 Let G be a unicyclic graph of order n and C l be the unique cycle of G.

Let G be a unicyclic graph, and let (C_{l}) be the unique cycle of G of length l.

Let G σ ∈ B 1 σ ( 2 k ) and C l be the unique cycle of G. Then l ≡ 0 ( mod 4 ).

Case 2. All pendent vertices are contained in the longest path of G σ. Let, in G σ, C r ( r < n ) be the unique cycle and P ( G σ ) = v 0 v 1 ⋯ v k be the longest path.

Let G σ ∈ U σ ( 2 k ) and C l be the unique cycle of G. Then we have: (1) If l is odd, then a 2 i ( G σ ) = m ( G, i ).   (2) If l is even and C l is oddly oriented, then a 2 i ( G σ ) = m ( G, i ) + 2 m ( G − C l, i − l 2 ).

For any unicyclic graph G, we assume that (C_{k}=v_{1}v_{2} cdots v_{k}v_{1}) is the unique cycle in G (for some k) and G has the form (U R_{1},ldots, R_{k})).

Then G − ≻ G +. Proof Let C l be the unique even cycle of G with length l.

Note that the arc ((k,j)) is a part of the unique cycle CQ of Q, and that the same unicyclic graph Q can be formed when each arc of CQ is added to a corresponding rooted tree T. Therefore, the double sum in (G_{2}) can be reorganized as a sum over all unicyclic subgraphs Q containing vertices ({1,2,ldots,m}).

Let (mathbb{U}(n)) be the set of n-vertex unicyclic graphs, each of whose vertices on the unique cycle is of degree at least three.

In this paper, we focus on a type of unicyclic graphs, each of whose vertices on the unique cycle is of degree at least three, and establish some bounds for their spectral radii, least eigenvalues, and the spreads.