The condition for a sequence to be potentially $A_{L‎, ‎M}$‎- graphic

Document Type : Research Paper

Authors

University of Kashmir

Abstract

The set of all non-increasing non-negative integer sequences $\pi=(d_1‎, ‎d_2,\ldots,d_n)$ is denoted by $NS_n$‎. ‎A sequence $\pi\in NS_{n}$ is said to be graphic if it is the degree sequence of a simple graph $G$ on $n$ vertices‎, ‎and such a graph $G$ is called a realization of $\pi$‎. ‎The set of all graphic sequences in $NS_{n}$ is denoted by $GS_{n}$‎. ‎The complete product split graph on $L‎ + ‎M$ vertices is denoted by $\overline{S}_{L‎, ‎M}=K_{L} \vee \overline{K}_{M}$‎, ‎where $K_{L}$ and $K_{M}$ are complete graphs respectively on $L = \sum\limits_{i = 1}^{p}r_{i}$ and $M = \sum\limits_{i = 1}^{p}s_{i}$ vertices with $r_{i}$ and $s_{i}$ being integers‎. ‎Another split graph is denoted by $S_{L‎, ‎M} = \overline{S}_{r_{1}‎, ‎s_{1}} \vee\overline{S}_{r_{2}‎, ‎s_{2}} \vee \cdots \vee \overline{S}_{r_{p}‎, ‎s_{p}}= (K_{r_{1}} \vee \overline{K}_{s_{1}})\vee (K_{r_{2}} \vee \overline{K}_{s_{2}})\vee \cdots \vee (K_{r_{p}} \vee \overline{K}_{s_{p}})$‎. ‎A sequence $\pi=(d_{1}‎, ‎d_{2},\ldots,d_{n})$ is said to be potentially $S_{L‎, ‎M}$-graphic (respectively $\overline{S}_{L‎, ‎M}$)-graphic if there is a realization $G$ of $\pi$ containing $S_{L‎, ‎M}$ (respectively $\overline{S}_{L‎, ‎M}$) as a subgraph‎. ‎If $\pi$ has a realization $G$ containing $S_{L‎, ‎M}$ on those vertices having degrees $d_{1}‎, ‎d_{2},\ldots,d_{L+M}$‎, ‎then $\pi$ is potentially $A_{L‎, ‎M}$-graphic‎. ‎A non-increasing sequence of non-negative integers $\pi = (d_{1}‎, ‎d_{2},\ldots,d_{n})$ is potentially $A_{L‎, ‎M}$-graphic if and only if it is potentially $S_{L‎, ‎M}$-graphic‎. ‎In this paper‎, ‎we obtain the sufficient condition for a graphic sequence to be potentially $A_{L‎, ‎M}$-graphic and this result is a generalization of that given by J‎. ‎H‎. ‎Yin on split graphs‎.

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References

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