The number of spanning trees in a k5 chain graph
Zunaira Kosar, Shahid Zaman, Wajid Ali, Asad Ullah
Abstract
Abstract Spanning trees (ST) have numerous applications in various fields, including computer science, graph theory, network design, and optimization. ST uses in designing efficient network topologies. In computer networks, a ST ensures that all nodes are connected while avoiding loops and redundant connections. This helps in minimizing network congestion and ensuring reliable communication. In routing protocols the ST’s play a crucial role such as the Spanning Tree Protocol (STP) and Rapid Spanning Tree Protocol (RSTP). These protocols are used to prevent network loops and establish a loop-free topology for efficient packet forwarding. In image processing the ST’s have applications in algorithms such as image segmentation and region merging. By constructing a minimum spanning tree based on pixel intensities or other image features, regions or objects can be efficiently grouped together. In wireless sensor networks, ST’s are employed to organize the network structure, reduce energy consumption, and enable efficient data collection. By constructing a spanning tree, sensor nodes can communicate with a central node or sink node while minimizing energy consumption and prolonging network lifetime. Motivated by the above applications and by the normalized Laplacian (NL) matrix, we obtained the spectrums of the n copies of K 5 chain graph, and is denoted by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>5</mml:mn> </mml:mrow> <mml:mrow> <mml:mi mathvariant="italic">ℓ</mml:mi> </mml:mrow> </mml:msubsup> </mml:math> . By utilizing these spectrums, we have successfully calculated the number of spanning trees for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>5</mml:mn> </mml:mrow> <mml:mrow> <mml:mi mathvariant="italic">ℓ</mml:mi> </mml:mrow> </mml:msubsup> </mml:math> .