The displacement and split decompositions for a $Q$-polynomial distance-regular graph

On the Sombor index and Sombor energy of m-splitting graph and m-shadow graph of regular graphs

A vertex-degree-based topological index named as Sombor index of a simple graph G with n vertices was recently introduced by I. Gutman. In this paper, we find Sombor index of m-splitting graph and m-shadow graph. Also, we determine relation between energy and Sombor energy of m-splitting graph and m-shadow graph of k-regular graph.

A duality between pairs of split decompositions for a $Q$-polynomial distance-regular graph

Let $Γ$ denote a $Q$-polynomial distance-regular graph with diameter $D \geq 3$ and standard module $V$. Recently Ito and Terwilliger introduced four direct sum decompositions of $V$; we call these the $(μ,ν)$--{\it split decompositions} of $V$, where $μ, ν\in \lbrace \downarrow, \uparrow \rbrace$. In this paper we show that the ($\downarrow,\downarrow$)--split decomposition and the ($\uparrow,\uparrow$)--split decomposition are dual with respect to the standard Hermitian form on $V$. We also show that the ($\downarrow,\uparrow$)--split decomposition and the ($\uparrow,\downarrow$)--split deco…

Splitting cubic circle graphs

We show that every 3-regular circle graph has at least two pairs of twin vertices; consequently no such graph is prime with respect to the split decomposition. We also deduce that up to isomorphism, K_4 and K_{3,3} are the only 3-connected, 3-regular circle graphs.

分裂亚循环$2$-群上的正则$t$-平衡Cayley地图

Regular $t$-balanced Cayley maps on split metacyclic $2$-groups

本文研究群上的正则$t$-平衡Cayley地图，并完全分类了分裂亚循环$2$-群上的此类地图。

On sum edge-coloring of regular, bipartite and split graphs

An edge-coloring of a graph $G$ with natural numbers is called a sum edge-coloring if the colors of edges incident to any vertex of $G$ are distinct and the sum of the colors of the edges of $G$ is minimum. The edge-chromatic sum of a graph $G$ is the sum of the colors of edges in a sum edge-coloring of $G$. It is known that the problem of finding the edge-chromatic sum of an $r$-regular ($r\geq 3$) graph is $NP$-complete. In this paper we give a polynomial time $(1+\frac{2r}{(r+1)^{2}})$-approximation algorithm for the edge-chromatic sum problem on $r$-regular graphs for $r\geq 3$. Also, it i…