Thin Distance-Regular Graphs with Classical Parameters $(D,q,q, \frac{q^{t}-1}{q-1}-1)$ with $t> D$ are the Grassmann Graphs

In the survey paper by Van Dam, Koolen and Tanaka (2016), they asked to classify the thin $Q$-polynomial distance-regular graphs. In this paper, we show that a thin distance-regular graph with the same intersection numbers as a Grassmann graph $J_q(n, D)~ (n \geqslant 2D)$ is the Grassmann graph if $D$ is large enough.

THE STRUCTURE OF GRAPHS ON n VERTICES WITH THE DEGREE SUM OF ANY TWO NONADJACENT VERTICES EQUAL TO n-2

Let G be an undirected simple graph on n vertices and sigma2(G)=n-2 (degree sum of any two non-adjacent vertices in G is equal to n-2) and alpha(G) be the cardinality of an maximum independent set of G. In G, a vertex of degree (n-1) is called total vertex. We show that, for n>=3 is an odd number then alpha(G)=2 and G is a disconnected graph; for n>=4 is an even number then 2=<alpha(G)<=(n+2)/2, where if alpha(G)=2 then G is a disconnected graph, otherwise G is a connected graph, G contains k total vertices and n-k vertices of degree delta=(n-2)/2, where 0<=k<=(n-2)/2. In particular, when k=0 then G is an (n-2)/2-Regular graph.

On Locating-Dominating Set of Regular Graphs

Let G be a simple, connected, and finite graph. For every vertex v ∈ V G , we denote by N G v the set of neighbours of v in G . The locating-dominating number of a graph G is defined as the minimum cardinality of W ⊆ V G such that every two distinct vertices u , v ∈ V G \ W satisfies ∅ ≠ N G u ∩ W ≠ N G v ∩ W ≠ ∅ . A graph G is called k -regular graph if every vertex of G is adjacent to k other vertices of G . In this paper, we determine the locating-dominating number of k -regular graph of order n , where k = n − 2 or k = n − 3 .

Some Chemistry Indices of Clique-Inserted Graph of a Strongly Regular Graph

In this paper, we give the relation between the spectrum of strongly regular graph and its clique-inserted graph. The Laplacian spectrum and the signless Laplacian spectrum of clique-inserted graph of strongly regular graph are calculated. We also give formulae expressing the energy, Kirchoff index, and the number of spanning trees of clique-inserted graph of a strongly regular graph. And, clique-inserted graph of the triangular graph T t , which is a strongly regular graph, is enumerated.

Distance-Regular Graph with Iintersection Array {140,108,18;1,18,105} Does not Exist

Графом Шилла называется дистанционно регулярный граф $\Gamma$ диаметра 3, имеющий второе собственное значение $\theta_1$, равное $a=a_3$. В этом случае $a$ делит $k$ и полагают $b=b(\Gamma)=k/a$. Юришич и Видали нашли массивы пересечений $Q$-полиномиальных графов Шилла с $b_2=c_2$: $\{2rt(2r+1),(2r-1)(2rt+t+1),r(r+t);1,r(r+t),t(4r^2-1)\}$. Однако многие массивы из этой серии не являются допустимыми. Белоусов И. Н. и Махнев А. А. нашли новую бесконечную серию допустимых массивов пересечений $Q$-полиномиальных графов Шилла с $b_2=c_2$ ($t=2r^2-1$): $\{2r(2r^2-1)(2r+1),(2r-1)(2r(2r^2-1)+2r^2),r(2r^2+r-1);1,r(2r^2+r-1),(2r^2-1)(4r^2-1)\}$. При $r=2$ получим массив пересечений $\{140,108,18;1,18,105\}$. В работе доказано, что граф с таким массивом пересечений не существует.