Distance-regular graph with large a1 or c2

In this paper, we study distance-regular graphs $\Gamma$ that have a pair of distinct vertices, say x and y, such that the number of common neighbors of x and y is about half the valency of $\Gamma$. We show that if the diameter is at least three, then such a graph, besides a finite number of exceptions, is a Taylor graph, bipartite with diameter three or a line graph.Comment: We submited this manuscript to JCT

Line graphs and 22-geodesic transitivity

For a graph $\Gamma$, a positive integer $s$ and a subgroup G\leq \Aut(\Gamma), we prove that $G$ is transitive on the set of $s$-arcs of $\Gamma$ if and only if $\Gamma$ has girth at least $2(s-1)$ and $G$ is transitive on the set of $(s-1)$-geodesics of its line graph. As applications, we first prove that the only non-complete locally cyclic $2$-geodesic transitive graphs are the complete multipartite graph $K_{3[2]}$ and the icosahedron. Secondly we classify 2-geodesic transitive graphs of valency 4 and girth 3, and determine which of them are geodesic transitive

Circumferences of 3-connected claw-free graphs, II

For a graph H , the circumference of H , denoted by c ( H ) , is the length of a longest cycle in H . It is proved in Chen (2016) that if H is a 3-connected claw-free graph of order n with δ ≥ 8 , then c ( H ) ≥ min { 9 δ − 3 , n } . In Li (2006), Li conjectured that every 3-connected k -regular claw-free graph H of order n has c ( H ) ≥ min { 10 k − 4 , n } . Later, Li posed an open problem in Li (2008): how long is the best possible circumference for a 3-connected regular claw-free graph? In this paper, we study the circumference of 3-connected claw-free graphs without the restriction on regularity and provide a solution to the conjecture and the open problem above. We determine five families F i ( 1 ≤ i ≤ 5 ) of 3-connected claw-free graphs which are characterized by graphs contractible to the Petersen graph and show that if H is a 3-connected claw-free graph of order n with δ ≥ 16 , then one of the following holds: (a) either c ( H ) ≥ min { 10 δ − 3 , n } or H ∈ F 1 . (b) either c ( H ) ≥ min { 11 δ − 7 , n } or H ∈ F 1 ∪ F 2 . (c) either c ( H ) ≥ min { 11 δ − 3 , n } or H ∈ F 1 ∪ F 2 ∪ F 3 . (d) either c ( H ) ≥ min { 12 δ − 10 , n } or H ∈ F 1 ∪ F 2 ∪ F 3 ∪ F 4 . (e) if δ ≥ 23 then either c ( H ) ≥ min { 12 δ − 7 , n } or H ∈ F 1 ∪ F 2 ∪ F 3 ∪ F 4 ∪ F 5 . This is also an improvement of the prior results in Chen (2016), Lai et al. (2016), Li et al. (2009) and Mathews and Sumner (1985)

Degree and neighborhood conditions for hamiltonicity of claw-free graphs

For a graph H , let σ t ( H ) = min { Σ i = 1 t d H ( v i ) | { v 1 , v 2 , … , v t } is an independent set in H } and let U t ( H ) = min { | ⋃ i = 1 t N H ( v i ) | | { v 1 , v 2 , ⋯ , v t } is an independent set in H } . We show that for a given number ϵ and given integers p ≥ t \u3e 0 , k ∈ { 2 , 3 } and N = N ( p , ϵ ) , if H is a k -connected claw-free graph of order n \u3e N with δ ( H ) ≥ 3 and its Ryjác̆ek’s closure c l ( H ) = L ( G ) , and if d t ( H ) ≥ t ( n + ϵ ) ∕ p where d t ( H ) ∈ { σ t ( H ) , U t ( H ) } , then either H is Hamiltonian or G , the preimage of L ( G ) , can be contracted to a k -edge-connected K 3 -free graph of order at most max { 4 p − 5 , 2 p + 1 } and without spanning closed trails. As applications, we prove the following for such graphs H of order n with n sufficiently large: (i) If k = 2 , δ ( H ) ≥ 3 , and for a given t ( 1 ≤ t ≤ 4 ), then either H is Hamiltonian or c l ( H ) = L ( G ) where G is a graph obtained from K 2 , 3 by replacing each of the degree 2 vertices by a K 1 , s ( s ≥ 1 ). When t = 4 and d t ( H ) = σ 4 ( H ) , this proves a conjecture in Frydrych (2001). (ii) If k = 3 , δ ( H ) ≥ 24 , and for a given t ( 1 ≤ t ≤ 10 ) d t ( H ) \u3e t ( n + 5 ) 10 , then H is Hamiltonian. These bounds on d t ( H ) in (i) and (ii) are sharp. It unifies and improves several prior results on conditions involved σ t and U t for the hamiltonicity of claw-free graphs. Since the number of graphs of orders at most max { 4 p − 5 , 2 p + 1 } are fixed for given p , improvements to (i) or (ii) by increasing the value of p are possible with the help of a computer

Shilla distance-regular graphs

A Shilla distance-regular graph G (say with valency k) is a distance-regular graph with diameter 3 such that its second largest eigenvalue equals to a3. We will show that a3 divides k for a Shilla distance-regular graph G, and for G we define b=b(G):=k/a3. In this paper we will show that there are finitely many Shilla distance-regular graphs G with fixed b(G)>=2. Also, we will classify Shilla distance-regular graphs with b(G)=2 and b(G)=3. Furthermore, we will give a new existence condition for distance-regular graphs, in general.Comment: 14 page

On some intriguing problems in Hamiltonian graph theory -- A survey

We survey results and open problems in Hamiltonian graph theory centred around three themes: regular graphs, $t$-tough graphs, and claw-free graphs