The Cartesian product of graphs G and H is the graph <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="34_2025_3124_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$G \Box H$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>□</mo> <mi>H</mi> </mrow> </math></EquationSource> </InlineEquation> whose vertex set is <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="34_2025_3124_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="103" /> </InlineMediaObject> <EquationSource Format="TEX">$V (G)\times V (H)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and whose edge set is the set of all pairs <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="34_2025_3124_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="102" /> </InlineMediaObject> <EquationSource Format="TEX">$(u_1,v_1)(u_2,v_2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>v</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>v</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> such that either <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="34_2025_3124_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="94" /> </InlineMediaObject> <EquationSource Format="TEX">$u_1u_2 \in E(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> <msub> <mi>u</mi> <mn>2</mn> </msub> <mo>∈</mo> <mi>E</mi> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="34_2025_3124_Article_IEq5.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">$v_1 = v_2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>v</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>v</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>, or <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="34_2025_3124_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="93" /> </InlineMediaObject> <EquationSource Format="TEX">$v_1v_2 \in E(H)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>v</mi> <mn>1</mn> </msub> <msub> <mi>v</mi> <mn>2</mn> </msub> <mo>∈</mo> <mi>E</mi> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="34_2025_3124_Article_IEq7.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">$u_1 = u_2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>u</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>. The multiple extended complete split-like graph <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="34_2025_3124_Article_IEq8.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="72" /> </InlineMediaObject> <EquationSource Format="TEX">$MECS_{b,s}^{a}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <mi>E</mi> <mi>C</mi> <msubsup> <mi>S</mi> <mrow> <mi>b</mi> <mo>,</mo> <mi>s</mi> </mrow> <mi>a</mi> </msubsup> </mrow> </math></EquationSource> </InlineEquation> is the join graph of an empty graph <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="34_2025_3124_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">${\overline{K}}_a$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mover> <mi>K</mi> <mo>¯</mo> </mover> <mi>a</mi> </msub> </math></EquationSource> </InlineEquation> and s copies of <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="34_2025_3124_Article_IEq10.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">$K_b\Box K_2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>K</mi> <mi>b</mi> </msub> <mo>□</mo> <msub> <mi>K</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="34_2025_3124_Article_IEq10.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">$K_b\Box K_2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>K</mi> <mi>b</mi> </msub> <mo>□</mo> <msub> <mi>K</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> is the Cartesian product of complete graphs <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="34_2025_3124_Article_IEq12.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$K_b$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mi>b</mi> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="34_2025_3124_Article_IEq13.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$K_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>. In this paper, we derive the resistance distance and the number of spanning trees of multiple extended complete split-like graph.

Resistance Distance and Spanning Trees of Multiple Extended Complete Split-Like Graph

期刊论文
发表日期：

2025年5月1日

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Chenlin Yang,
Tao Tian

摘要

The Cartesian product of graphs G and H is the graph $G \Box H$ $G □ H$ whose vertex set is $V (G)\times V (H)$ $V (G) \times V (H)$ and whose edge set is the set of all pairs $(u_1,v_1)(u_2,v_2)$ $(u_{1}, v_{1}) (u_{2}, v_{2})$ such that either $u_1u_2 \in E(G)$ $u_{1} u_{2} \in E (G)$ and $v_1 = v_2$ $v_{1} = v_{2}$ , or $v_1v_2 \in E(H)$ $v_{1} v_{2} \in E (H)$ and $u_1 = u_2$ $u_{1} = u_{2}$ . The multiple extended complete split-like graph $MECS_{b,s}^{a}$ $M E C S_{b, s}^{a}$ is the join graph of an empty graph ${\overline{K}}_a$ ${\bar{K}}_{a}$ and s copies of $K_b\Box K_2$ $K_{b} □ K_{2}$ , where $K_b\Box K_2$ $K_{b} □ K_{2}$ is the Cartesian product of complete graphs $K_b$ $K_{b}$ and $K_2$ $K_{2}$ . In this paper, we derive the resistance distance and the number of spanning trees of multiple extended complete split-like graph.