Time trickled by with the turning of pages.
The sun crept from the eastern windows to directly overhead, brightening the reading room.
At noon, they each went to the cafeteria for lunch. Upon returning, they resumed their places at opposite ends of the long table, not disturbing one another.
The afternoon air grew stuffier, and even the ceiling fan overhead seemed to struggle as it turned.
Chen Zhuo picked up the latest issue of *Discrete Mathematics* from the cart. It had a deep blue cover and felt substantial in his hands.
He opened it to the table of contents and gave it a cursory glance, his eyes landing on an article with a very long title.
The article was about the problem of proving the lower bound for a certain class of bipartite graphs.
Chen Zhuo had encountered this problem before while reading older papers. It was a classic tough nut to crack in combinatorial mathematics. Many mathematicians had tried to raise the value of this lower bound, but a general proof path remained elusive.
He flipped to the article’s page.
It was a long article, a sprawling thirty-plus pages.
The author was a professor at a university in the United Kingdom. Chen Zhuo settled his mind and began reading from the introduction in the first section.
The author’s approach was very traditional, very orthodox.
To prove the lower bound, he used a pure combinatorial construction method. The article defined a large number of subgraph structures, then pieced them together one by one, like a jigsaw puzzle.
With each piece added, a lemma was needed to prove that the join was logically sound and didn’t violate the original graph-theoretical properties.
Chen Zhuo looked at the densely packed pages of subgraph classifications and constraints.
Case one: assume the vertex degree is greater than a certain value.
Case two: assume a specific cycle exists.
Case three...
The author’s writing was exceptionally rigorous.
Every step of his deduction was correct, every proof for every lemma unassailable. He was like an extremely patient bricklayer, using bricks and mortar to build this wall, bit by bit, layer by layer.
There were no shortcuts, just sheer, head-on grit.
After finishing the section, Chen Zhuo leaned back in his chair and rubbed the bridge of his nose.
In academia, an article that so thoroughly and meticulously pinned down a problem using an exhaustive construction method was absolutely qualified for publication in a top-tier journal.
But as he followed the author’s train of thought, a different picture uncontrollably surfaced in his mind.
For the past few days, his mind had been filled with the tools of algebraic matrices.
Looking at the complex graphs that had been broken down into dozens of cases for discussion in geometric space, an idea suddenly occurred to him.
’What is the essence of a graph?’
’It’s a set of vertices, and the connections between them.’
’What if these intricate connections were abstracted directly into an adjacency matrix of zeros and ones?’
’Once the graph becomes a matrix, what about all those graph-theoretical properties discussed repeatedly in these thirty pages?’
’Properties like connectivity, bipartiteness, and even that troublesome lower bound value...’
’Don’t they just become a problem of finding the matrix’s eigenvalues?’
A faint light sparked in Chen Zhuo’s eyes.
It wasn’t that he thought he was smarter than the professor. It was just that this summer, he happened to have trained his thinking in discrete algebra to an almost instinctual level.
The professor was looking at this problem from a purely combinatorial mathematics perspective, which was why he could only piece the puzzle together one block at a time.
But Chen Zhuo now had an interdisciplinary measuring stick in his hands.
He sat up straight again, pushed the copy of *Discrete Mathematics* aside, and took out a clean sheet of A4 paper.
He just wanted to give it a try.
To see if he could use the tools of algebra to simplify this cumbersome construction process, even just a little.
He picked up a black gel pen, wrote a basic graph theory definition at the top of the paper, and then drew a corresponding matrix directly below it.
The tip of the pen met the paper, making a soft, even scratching sound.
Chen Zhuo wrote with intense focus.
He didn’t even notice when Su Wei walked over to his table.
This afternoon’s derivations, however, were not as straightforward as one might imagine.
Forcibly translating a purely combinatorial graph problem into the dimension of algebraic matrices, the initial mapping was indeed smooth.
Connectivity, which would have required long paragraphs of text to describe, was easily packed into a symmetric matrix.
But this was just the beginning.
In that thirty-plus-page paper, the original author had listed an extremely complex set of boundary conditions to prove the lower bound.
If Chen Zhuo wanted to compress all these conditions losslessly into the range of the matrix’s eigenvalues, he would need to construct several very clever inequalities to handle the scaling.
That wasn’t something that could be conjured out of thin air in the blink of an eye.
It required time for repeated comparisons, for trying out different algebraic tools.
Chen Zhuo’s pen hovered in mid-air, pausing before the value range of an eigenvalue.
He was slowly building a scaffold in his mind.
Su Wei had originally gone to the water dispenser at the end of the hallway to wash her cup. On her way back, she glanced over at Chen Zhuo.
For the past few days, Chen Zhuo had been reading very quickly, usually flipping through a few pages, jotting down a line in his notebook, and then flipping again. But this afternoon, he had been in a writing posture at his desk for nearly an hour.
She paused, her gaze falling on the scratch paper under Chen Zhuo’s hand.
Instead of the usual scattered sentences, the paper was filled with line after line of meticulously arranged matrix derivations.

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